KYIV SCHOOL OF ECONOMICS
Financial Econometrics (2nd part):
Introduction to Financial Time Series
May 2011
Instructor: Maksym Obrizan
Lecture notes I
# 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?
# 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
# 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
# 7. Basic concepts
Let
The sample mean is
# 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
# 8. The sample variance is
# 9. The third central moment measures the
symmetry of X with respect to its mean
(skewness)
# 10.
0.4
Skewed to the left
Standard normal
0.35
0.3
0.25
0.2
0.15
0.1
For figure on slide 10: Skewness is 0.007 for
standard normal and -0.839 for skewed to
the left
0.05
0
-25
-20
-15
-10
-5
0
5
0.4
4th
# 11. The
central moment measures the tail
behavior of X
# 12.
t with 1 df
Standard normal
0.35
0.3
0.25
0.2
Excess kurtosis K(x)-3 :
0.15
0.1
0.05
Kurtosis of Student t distrbution with 1 df is 3.43
on slide 12
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
# 13. Skewness and Kurtosis in returns data
# 14. Stationarity –
Skewness
A time series {rt} is strictly stationary if
High excess kurtosis
A time series {rt} is weakly stationary if
In practice, high excess kurtosis means that the
distribution of returns tends to contain more
extreme values than the standard normal
In this course: weakly stationary time series
# 15. Linear time series
# 16. Quick review of autoregressive models
AR(p) model is
The mean
and the variance
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)
# 18. The estimate
of the second
equation is called the lag-2 sample PACF of
rt .
Partial Autocorrelation Function (PACF):
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
# 23. Parameter Estimation
# 22. NOTES
# 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
Sample Autocorrelation Function (ACF)
# 26.
0.8
Sample Autocorrelation
# 25. After you fit the model obtain residual
series to check for remaining autocorrelation
Suggestion:
Also plot 95% confidence intervals
0.6
0.4
0.2
0
Graph to the left?
-0.2
Graph below?
0
2
4
6
8
10
Lag
12
14
16
Sample Autocorrelation Function (ACF)
# 28. Ljung-Box (1978) statistics –
# 27.
Sample Autocorrelation
0.8
0.6
0.4
0.2
In practice, the choice of m may affect the
performance of Q(m) statistics
Simulations suggest to set m to approx. ln(T)
0
-0.2
0
2
4
6
8
10
Lag
12
14
16
18
20
18
20
# 29. For an AR(p) model, the Ljung-Box
statistics Q(m) follows asymptotically a chisquared 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
# 32. Multistep Ahead Forecast
Forecast often employs the minimum squared
error term loss function
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
# 36. This equation can be re-written as the
second-order difference equation
where B is called back-shift operator such that
It can be shown that the ACF of a stationary
AR(2) model satisfies
Sometimes lag operator L is used instead of B
# 37. Corresponding to the difference equation
there is quadratic equation
# 38. If characteristic roots are complex numbers
(complex conjugate pair) then the ACF of
this series shows damping sine and cosine
waves
which can be solved for characteristic roots
For example, AR(2) model
Interesting case when
The graph is in the bottom left corner
Sample Autocorrelation Function (ACF)
1
# 39.
# 40. In business and economic applications
complex characteristic roots give rise to
business cycles
0.8
Sample Autocorrelation
0.6
For an AR(2) model on slide # 35 with a pair of
complex characteristic roots the average
length of the stochastic cycles is
0.4
0.2
0
-0.2
-0.4
-0.6
where the cosine inverse is stated in degrees
0
5
10
15
20
Lag
25
30
35
40
# 41. Illustration: US GNP seasonally adjusted
from QII.1947 to Q1.1991
Fit AR(3) model
Obtain a corresponding third-order difference
equation
# 43. Application: Seasonal Models
Quarterly earnings per share of a company may
exhibit cyclical or periodic behavior –
seasonal time series
# 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
# 44.
# 45. Seasonal differencing
# 46.
In general, for a time series with periodicity s:
# 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