金融时间序列分析

Lecture Notes of Bus41202(Spring2010)

Analysis of Financial Time Series

Ruey S.Tsay

Simple AR models:(Regression with lagged variables.) Motivating example:The growth rate of U.S.quarterly real GNP from1947to1991.Recall that the model discussed before is

r t=0.005+0.35r t−1+0.18r t−2−0.14r t−3+a t,ˆσa=0.01.

This is called an AR(3)model because the growth rate r t depends on the growth rates of the past three quarters.How do we specify this model from the data?Is it adequate for the data?What are the implications of the model?These are the questions we shall address in this lecture.

Another example:U.S.monthly unemployment rate.

AR(1)model:

1.Form:r t=φ0+φ1r t−1+a t,whereφ0andφ1are real numbers,

which are referred to as“parameters”(to be estimated from the data in an application).For example,

r t=0.005+0.2r t−1+a t

2.Stationarity:necessary and sufficient condition|φ1|<1.Why?

3.Mean:E(r t)=φ0

1−φ1

Time

g n p

19501960197019801990

−0.02−0.010.000.010.020.03

0.0

4U.S. quarterly real GNP growth rate: 1947.II to 1991.I

Figure 1:U.S.quarterly growth rate of real GNP:1947-1991

4.Alternative representation:Let E (r t )=µbe the mean of r t so that µ=φ0/(1−φ1).Equivalently,φ0=µ(1−φ1).Plugging in the model,we have

(r t −µ)=φ1(r t −1−µ)+a t .

(1)

This model also has two parameters (µand φ1).It explicitly uses the mean of the series.It is less commonly used in the literature,but is the model representation used in R.5.Variance:Var(r t )=

σ2a 1−φ21

.6.Autocorrelations:ρ1=φ1,ρ2=φ21,etc.In general,ρk =φk

1

and ACF ρk decays exponentially as k increases,

Time

y

1950

196019701980199020002010

4

6

81

U.S. monthly unemployment rate 1948.1 to 2010.2

Figure 2:U.S.monthly unemployment rate (total civilian,16and older)from January 1948to February 2010

7.Forecast(minimum squared error):Suppose the forecast origin is n.For simplicity,we shall use the model representation in(1) and write x t=r t−µ.The model then becomes x t=φ1x t−1+a t. Note that forecast of r t is simply the forecast of x t plusµ.

(a)1-step ahead forecast at time n:

ˆx n(1)=φ1x n

(b)1-step ahead forecast error:

e n(1)=x n+1−ˆx n(1)=a n+1

Thus,a n+1is the un-predictable part of x n+1.It is the shock at time n+1!

(c)Variance of1-step ahead forecast error:

Var[e n(1)]=Var(a n+1)=σ2a.

(d)2-step ahead forecast:

ˆx n(2)=φ1ˆx n(1)=φ21x n.

(e)2-step ahead forecast error:

e n(2)=x n+2−ˆx n(2)=a n+2+φ1a n+1

(f)Variance of2-step ahead forecast error:

Var[e n(2)]=(1+φ21)σ2a

which is greater than or equal to Var[e n(1)],implying that uncertainty in forecasts increases as the number of steps in-creases.