计量经济学 美国本科Lecture 9 Time Series

Autocorrelation
Applying this to a time series data {Yt }, the j th autocovariance of a series Yt is the covariance between Yt and its j th lag, Yt −j . The j th autocovariance, denoted by γj : γj = cov (yt , yt −j )
White Noise: An Example
Example: 2 A White Noise is a process { t } such that E ( t ) = 0, E ( 2 t) = σ , and E ( t , t −j ) = 0 for any j = 0. (Stationary)
The j th autocorrelation, denoted by ρj : ρj ≡ corr (yt , yt −j ) = cov (yt , yt −j ) var (yt )var (yt −j )
Stationarity
Definition: Consider a sequence of time series {Yt } with mean µt and covariance γjt . If both µt and γjt do not depend on the date t, then the process for {Yt } is covariance-stationary or weakly stationary: E (Yt ) = µ E (Yt − µt )(Yt −j − µt −j ) = γj < ∞ What does stationary mean in practice? Past: time plot of Yt varies around a fixed level within a finite range! Future: the mean and variance of future Yt are the same as those of the data so that meaningful inferences can be made.
Aggregate consumption and GDP for a country (for example, 20 years of quarterly observations = 80 observations). U.S. unemployment rate, annualized inflation rate.
Autocorrelation
In time series data, the value of Y in one period is typically correlated with its value in the next period. The correlation of a series with its own lagged value is called autocorrelation or serial correlation. To recall, the covariance between two random variables are defined as: cov (xi , yi ) = E {(xi − µx )(yi − µy )} The correlation is defined as: corr (xi , yi ) = cov (xi , yi ) var (xi )var (yi )
AR(q)
Variance of AR(1): Var (Yt ) = σ 2 /(1 − φ2 ) Proof: Take variance on both hand sides: LHS = Var (Yt ) RHS = Var (c + φYt −1 + t ) = φ2 Var (Yt −1 )+ Var ( t )+ 2Cov (φYt −1 , t ) = φ2 Var (Yt ) + σ 2 ⇒ Var (Yt ) = σ 2 /(1 − φ2 ) A p th-order autoregressive process, denoted AR(p), is characterized by Yt = c + φ1 Yt −1 + φ2 Yt −2 + ... + φp Yt −p + t where c and φ1 , ..., φq are real constants, { t } is a white noise.
In other words, Li preceding yt means to lag yt by i periods. L raised to negative power is a lead operator: L−i yt ≡ yt +i
For example: Lyt = yt −1 L2 yt ≡ yt −2 L−1 yt ≡ yt +1
Lecture 9: Introduction to Time Series
Xing Hong
Department of Economics University of Maryland, College Park
Spring 2016
What is Time Series data?
Time series data, {yt , yt +1 , yt +2 , ...}, are data collected on the same observational unit at multiple time periods. For example:
is a white noise. Is Xt covariance
White Noise: An Example
Figure: White Noise
Notation: Lag Operator
The lag operator L is defined to be a linear operator such that for any value of yt Li yt ≡ yt −i
] = E[
2 t
+ 2θ
t t −1
+θ
2 t −1 ]
+ θE (
2 t −1 )
= σ2 + θ2 σ2
MA(q)
A q th-order moving average process, denoted MA(q ), is characterized by Yt = µ + t + θ1 t −1 + θ2 t −2 + ... + θq t −q where µ and θ1 , ..., θq are real constants, { t } is a white noise.
t
+θ
t −1 )
= E (µ) + E ( t ) + θ E (
t −1 )
=µ
Variance: Var (Yt ) = (1 + θ2 )σ 2 . Proof: Var (Yt ) = E [(Yt − µ)2 ] = E [( t + θ = E( 2 t ) + 2θ E (
t t −1 ) t −1 ) 2
An Example: Real Gross Domestic Product
Some Notation
Consider a time series {Yt , Yt +1 , Yt +2 , ...}, The first lag of a time series Yt is Yt −1 ; its j th lag is Yt −j . The first difference of a series, ∆Yt , is its change between periods t − 1 and t , that is ∆Yt = Yt − Yt −1 . The first difference of the logarithm of Yt is ∆ln(Yt ) = ln(Yt ) − ln(Yt −1 ). The percentage change of a time series Yt between periods t − 1 and t is approximately 100∆ln(Yt ).
Moving Average Processes
A first order moving average (MA) process, MA(1), is Yt = µ +
t
+θ
t −1
where µ and θ are real constants, { t } is a white noise.
Mean: E (Yt ) = µ. Proof: E (Yt ) = E (µ +
White noise with drift (Stationary)’: Yt = µ +
t
White noise with time drift (non-stationary): Yt = β t +
t
Exercise: Xt = a t + b stationary?
t −1 ,
where
t
Autoregressive Processes
A first order AutoRegressive process, AR(1), is Yt = c + φYt −1 +
t
where c and φ are real constants, { t } is a white noise. Remark: Need |φ| < 1 for {Yt } to be covariance-stationary. Assume |φ| < 1 we have the following properties: