Econometría 2: Análisis de series de Tiempo

Transcription

1 Econometría 2: Análisis de series de Tiempo Karoll GOMEZ Segundo semestre 2016

2 II. Basic definitions

3 A time series is a set of observations X t, each one being recorded at a specific time t with 0 < t < T. In reality we can only observe the time series at a finite number of times, and in that case the underlying sequence of random variables (X 1, X 2,..., X t ) is just a an t-dimensional random variable (or random vector), i.e. finite number of observations.

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5 Remark: The realization (the result or the observed value) of a random variable is a number. However, as it s a random variable, we know that the number can take values from a given set according to some probability law. The same applies to stochastic process, but now the realization instead of being a single number is a sequence (if the process is discrete) or a function (if it s continuous) of random variables. Basically, a time series.

6 DEFINITION 1: In that case X t, t = 1, 2,... is called a discrete stochastic process. In order to specify its statistical properties we then need to consider all t-dimensional distributions: P[X 1 x 1,..., X t x t ] t = 1, 2,... DEFINITION 2: A time series model for the observed data X t is a specification of the joint distributions (or possibly only the means and covariances) of a sequence of random variables x t of which X t is postulated to be a realization.

7 DEFINITION 3: A process X t, t Z is said to be an i.i.d noise with mean 0 and variance σ 2, written X t i.i.d(0, σ 2 ) if the random variables X t are independent and identically distributed with E[X t ] = 0 and Var(X t ) = σ 2 A stochastic process with T Z is often called a time series (Z denotes natural numbers).

8 DEFINITION 4: Let X t, t Z be a stochastic process with Var(X t ) <, the mean function of X t is: µ X (t) = E[X t ] t T the covariance function of X t is: γ X (r, s) = Cov[X r, X s ] r, s T

9 DEFINITION 5: The time series X t, t Z is said to be (weakly) stationary process if: Var(X (t)) < t T µ X (t) = µ t T γ X (r, s) = γ X (r + t, s + t) r, s T Loosely speaking, a stochastic process is stationary, if its statistical properties do not change with time. Notice also that the mean and variance must be finite.

10 DEFINITION 6: Let X t, t Z be a (weakly) stationary time series. The autocovariance function of X t is: γ X (τ) = Cov[X t, X t τ ] The autocorrelation function (ACF) of X t is: ρ X (τ) = γ X (τ) γ X (0) The value τ = r s is referred to as the lag.

11 NOTE: For non-stationary process: The autocovariance function of X t is: γ X (τ) = Cov[X t, X t τ ] The autocorrelation function (ACF) is: ρ X (τ) = Cov[X t, X t τ ] Var[Xt ] Var[X t τ ] The value τ = r s is referred to as the lag.

12 Remarks: 1. A series X t is said to be lagged if its time axis is shifted: shifting by k lags gives the series X t k. 2. A plot of ρ t against the lag k = 1, 2,..., m with m < T is called the correlogram 3. ρ t values are 1 < ρ t < 1 4. We use the sample to compute cov(x t, X t 1 ), cov(x t, X t 2 ),..., cov(x t, X t k )

13 Autocorrelation and autocorrelogram: more details

14 NOTE: c k and r k correspond to estimated sample values for γ(τ) and ρ(τ)

15 Interpreting autocorrelogram

16 Seasonal series

17 DEFINITION 7: The time series X t, t Z is said to be white noise process with with mean µ and variance σ 2, written: X t WN(µ, σ 2 ) if: E[X t ] = µ γ X (h) =σ 2 if h = 0 0 if h 0

18 DEFINITION 8: The time series X t, t Z is said to be (strictly) stationary process if the distribution of : (X t1,..., X tk ) and (X t1 +h,..., X tk +h) are the same for all set of data points t 1,..., t k and all h Z. Remarks: It means the joint distribution function is invariant under time shifts. Weak stationarity rely only on properties defined by the means and covariances, while strict stationarity rely only on all distribution properties.

19 A strict stationary process X t, t Z with Var(X t ) < is said to be stationary process. A stationary time series X t, t Z does not need to be strictly stationary. Example: X t is a sequence of independent variables and X t =EXP(1) if t is odd N(1, 1) if t is even Thus X t is WN(1, 1) but not i.i.d(1, 1).

20 DEFINITION 9: The time series X t, t Z is said to be Gaussian time series if the all finite dimensional distribution are normal. A stationary Gaussian time series X t, t Z is strictly stationary, since the normal distribution is determined by its mean and its covariance.

21 DEFINITION 10: Let B be the backward shift operator, i.e. (BX ) t = X t 1. In the obvious way we define powers of B (B j X ) t = X t j. The operator can be defined for linear combinations by B(c 1 X t1 + c 2 X t2 ) = c 1 X t1 1 + c 2 X t2 1 Also as (αb k + βb h )X t = αx t k + βx t h Strict stationarity means that B h X has the same distribution for all h Z.

22 DEFINITION 11: Let the differencing operator, defined by X t = (1 B)X t = X t X t 1 The power operator is defined as: 2 X t = ( X t ) = (X t X t 1 ) = (X t X t 1 ) (X t 1 X t 2 ) = X t 2X t 1 X t 2