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
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II. Basic definitions
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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.
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Modelos de series de tiempoDefinitions
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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.
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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.
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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).
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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
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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.
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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.
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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.
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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)
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Modelos de series de tiempoDefinitions
Autocorrelation and autocorrelogram: more details
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Modelos de series de tiempoDefinitions
NOTE: ck and rk correspond to estimated sample values for γ(τ) and ρ(τ)
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Modelos de series de tiempoDefinitions
Interpreting autocorrelogram
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Modelos de series de tiempoDefinitions
Seasonal series
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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
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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.
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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).
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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.
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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 .
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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