econometría 2: análisis de series de tiempo · modelos de series de tiempo de nitions remark: i...
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Econometrıa 2: Analisis de series de Tiempo
Karoll [email protected]
http://karollgomez.wordpress.com
Segundo semestre 2016
II. Basic definitions
Modelos de series de tiempoDefinitions
Definitions
I A time series is a set of observations Xt , each one beingrecorded at a specific time t with 0 < t < T .
I In reality we can only observe the time series at a finitenumber of times, and in that case the underlying sequence ofrandom variables (X1,X2, ...,Xt) is just a an t-dimensionalrandom variable (or random vector), i.e. finite number ofobservations.
Modelos de series de tiempoDefinitions
Modelos de series de tiempoDefinitions
Remark:
I The realization (the result or the observed value) of a randomvariable is a number.
I However, as it’s a random variable, we know that the numbercan take values from a given set according to some probabilitylaw.
I The same applies to stochastic process, but now therealization instead of being a single number is a sequence (ifthe process is discrete) or a function (if it’s continuous) ofrandom variables. Basically, a time series.
Modelos de series de tiempoDefinitions
DEFINITION 1: In that case Xt , t = 1, 2, ... is called a discretestochastic process. In order to specify its statistical properties wethen need to consider all t-dimensional distributions:
P[X1 6 x1, ...,Xt 6 xt ] ∀ t = 1, 2, ...
DEFINITION 2: A time series model for the observed data Xt isa specification of the joint distributions (or possibly only the meansand covariances) of a sequence of random variables xt of which Xt
is postulated to be a realization.
Modelos de series de tiempoDefinitions
DEFINITION 3: A process Xt , t ∈ Z is said to be an i.i.d noisewith mean 0 and variance σ2, written
Xt ∼ i .i .d(0, σ2)
if the random variables Xt are independent and identicallydistributed with E [Xt ] = 0 and Var(Xt) = σ2
A stochastic process with T ∈ Z is often called a time series (Zdenotes natural numbers).
Modelos de series de tiempoDefinitions
DEFINITION 4: Let Xt , t ∈ Z be a stochastic process withVar(Xt) <∞,the mean function of Xt is:
µX (t) = E [Xt ] t ∈ T
the covariance function of Xt is:
γX (r , s) = Cov [Xr ,Xs ] r , s ∈ T
Modelos de series de tiempoDefinitions
DEFINITION 5: The time series Xt , t ∈ Z is said to be (weakly)stationary process if:
Var(X (t)) <∞ ∀t ∈ T
µX (t) = µ t ∈ T
γX (r , s) = γX (r + t, s + t) r , s ∈ T
Loosely speaking, a stochastic process is stationary, if its statisticalproperties do not change with time.Notice also that the mean and variance must be finite.
Modelos de series de tiempoDefinitions
DEFINITION 6: Let Xt , t ∈ Z be a (weakly) stationary time series.The autocovariance function of Xt is:
γX (τ) = Cov [Xt ,Xt−τ ]
The autocorrelation function (ACF) of Xt is:
ρX (τ) =γX (τ)
γX (0)
The value τ = r − s is referred to as the lag.
Modelos de series de tiempoDefinitions
NOTE: For non-stationary process:
The autocovariance function of Xt is:
γX (τ) = Cov [Xt ,Xt−τ ]
The autocorrelation function (ACF) is:
ρX (τ) =Cov [Xt ,Xt−τ ]√
Var [Xt ]√
Var [Xt−τ ]
The value τ = r − s is referred to as the lag.
Modelos de series de tiempoDefinitions
Remarks:
1. A series Xt is said to be lagged if its time axis is shifted:shifting by k lags gives the series Xt−k .
2. A plot of ρt against the lag k = 1, 2, ...,m with m < T iscalled the correlogram
3. ρt values are −1 < ρt < 1
4. We use the sample to computecov(Xt ,Xt−1), cov(Xt ,Xt−2), ..., cov(Xt ,Xt−k)
Modelos de series de tiempoDefinitions
Autocorrelation and autocorrelogram: more details
Modelos de series de tiempoDefinitions
NOTE: ck and rk correspond to estimated sample values for γ(τ) and ρ(τ)
Modelos de series de tiempoDefinitions
Interpreting autocorrelogram
Modelos de series de tiempoDefinitions
Seasonal series
Modelos de series de tiempoDefinitions
DEFINITION 7: The time series Xt , t ∈ Z is said to be whitenoise process with with mean µ and variance σ2, written:
Xt ∼WN(µ, σ2)
if:E [Xt ] = µ
γX (h) =σ2 if h = 0
0 if h 6= 0
Modelos de series de tiempoDefinitions
DEFINITION 8: The time series Xt , t ∈ Z is said to be (strictly)stationary process if the distribution of :
(Xt1 , ...,Xtk ) and (Xt1+h, ...,Xtk+h)
are the same for all set of data points t1, ..., tk and all h ∈ Z .
Remarks:
I It means the joint distribution function is invariant under timeshifts.
I Weak stationarity rely only on properties defined by the meansand covariances, while strict stationarity rely only on alldistribution properties.
Modelos de series de tiempoDefinitions
I A strict stationary process Xt , t ∈ Z with Var(Xt) <∞ is saidto be stationary process.
I A stationary time series Xt , t ∈ Z does not need to bestrictly stationary.
Example: Xt is a sequence of independent variables and
Xt =EXP(1) if t is odd
N(1, 1) if t is even
Thus Xt is WN(1, 1) but not i .i .d(1, 1).
Modelos de series de tiempoDefinitions
DEFINITION 9: The time series Xt , t ∈ Z is said to be Gaussiantime series if the all finite dimensional distribution are normal.
I A stationary Gaussian time series Xt , t ∈ Z is strictlystationary, since the normal distribution is determined by itsmean and its covariance.
Modelos de series de tiempoDefinitions
DEFINITION 10: Let B be the backward shift operator, i.e.(BX )t = Xt−1.
I In the obvious way we define powers of B(B jX )t = Xt−j .
I The operator can be defined for linear combinations byB(c1Xt1 + c2Xt2) = c1Xt1−1 + c2Xt2−1
I Also as (αBk + βBh)Xt = αXt−k + βXt−h
I Strict stationarity means that BhX has the same distributionfor all h ∈ Z .
Modelos de series de tiempoDefinitions
DEFINITION 11: Let ∇ the differencing operator, defined by∇Xt = (1− B)Xt = Xt − Xt−1
The power operator is defined as:
∇2Xt = ∇(∇Xt)
= ∇(Xt − Xt−1)
= (Xt − Xt−1)− (Xt−1 − Xt−2)
= Xt − 2Xt−1 − Xt−2