Lecture 6 - Discrete Time Seriescr173/Sta444_Sp17/slides/Lec6.pdfWeakStationary...
Transcript of Lecture 6 - Discrete Time Seriescr173/Sta444_Sp17/slides/Lec6.pdfWeakStationary...
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Lecture 6Discrete Time Series
Colin Rundel02/06/2017
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Discrete Time Series
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Stationary Processes
A stocastic process (i.e. a time series) is considered to be strictly stationaryif the properties of the process are not changed by a shift in origin.
In the time series context this means that the joint distribution of{yt1 , . . . , ytn} must be identical to the distribution of {yt1+k, . . . , ytn+k} forany value of n and k.
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Stationary Processes
A stocastic process (i.e. a time series) is considered to be strictly stationaryif the properties of the process are not changed by a shift in origin.
In the time series context this means that the joint distribution of{yt1 , . . . , ytn} must be identical to the distribution of {yt1+k, . . . , ytn+k} forany value of n and k.
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Stationary Processes
A stocastic process (i.e. a time series) is considered to be strictly stationaryif the properties of the process are not changed by a shift in origin.
In the time series context this means that the joint distribution of{yt1 , . . . , ytn} must be identical to the distribution of {yt1+k, . . . , ytn+k} forany value of n and k.
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Weak Stationary
Strict stationary is too strong for most applications, so instead we often optfor weak stationary which requires the following,
1. The process has finite variance
E(y2t ) < ∞ for all t
2. The mean of the process in constant
E(yt) = µ for all t
3. The second moment only depends on the lag
Cov(yt, ys) = Cov(yt+k, ys+k) for all t, s, k
When we say stationary in class we almost always mean this version ofweakly stationary.
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Weak Stationary
Strict stationary is too strong for most applications, so instead we often optfor weak stationary which requires the following,
1. The process has finite variance
E(y2t ) < ∞ for all t
2. The mean of the process in constant
E(yt) = µ for all t
3. The second moment only depends on the lag
Cov(yt, ys) = Cov(yt+k, ys+k) for all t, s, k
When we say stationary in class we almost always mean this version ofweakly stationary.
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Autocorrelation
For a stationary time series, where E(yt) = µ and Var(yt) = σ2) for all t, wedefine the autocorrelation at lag k as
ρk = Cor(yt, yt+k)
=Cov(yt, yt+k)√Var(yt)Var(yt+k)
=E ((yt − µ)(yt+k − µ))
σ2
this is also sometimes written in terms of the autocovariance function (γk)as
γk = γ(t, t+ k) = Cov(yt, yt+k)
ρk =γ(t, t+ k)√
γ(t, t)γ(t+ k, t+ k)=
γ(k)γ(0)
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Autocorrelation
For a stationary time series, where E(yt) = µ and Var(yt) = σ2) for all t, wedefine the autocorrelation at lag k as
ρk = Cor(yt, yt+k)
=Cov(yt, yt+k)√Var(yt)Var(yt+k)
=E ((yt − µ)(yt+k − µ))
σ2
this is also sometimes written in terms of the autocovariance function (γk)as
γk = γ(t, t+ k) = Cov(yt, yt+k)
ρk =γ(t, t+ k)√
γ(t, t)γ(t+ k, t+ k)=
γ(k)γ(0)
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Covariance Structure
Based on our definition of a (weakly) stationary process, it implies acovariance of the following structure,
γ(0) γ(1) γ(2) γ(3) · · · γ(n)γ(1) γ(0) γ(1) γ(2) · · · γ(n − 1)γ(2) γ(1) γ(0) γ(1) · · · γ(n − 2)γ(3) γ(2) γ(1) γ(0) · · · γ(n − 3)...
......
.... . .
...γ(n) γ(n − 1) γ(n − 2) γ(n − 3) · · · γ(0)
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Example - Random walk
Let yt = yt−1 + wt with y0 = 0 and wt ∼ N (0, 1). Is yt stationary?
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t
y
Random walk
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ACF + PACF
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Series rw$y
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Lag
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CF
Series rw$y
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Example - Random walk with drift
Let yt = δ + yt−1 + wt with y0 = 0 and wt ∼ N (0, 1). Is yt stationary?
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Random walk with trend
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ACF + PACF
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Series rwt$y
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Example - Moving Average
Let wt ∼ N (0, 1) and yt = 13 (wt−1 + wt + wt+1), is yt stationary?
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Moving Average
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ACF + PACF
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Series ma$y
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Series ma$y
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Autoregression
Let wt ∼ N (0, 1) and yt = yt−1 − 0.9yt−2 + wt with yt = 0 for t < 1, is ytstationary?
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Autoregressive
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ACF + PACF
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Series ar$y
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Example - Australian Wine Sales
Australian total wine sales by wine makers in bottles
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Time series
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40000
1980 1985 1990 1995
date
sale
s
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Basic Model Fit
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1980 1985 1990 1995
date
sale
s
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Residuals
resid_q
resid_l
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resid_q
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Autocorrelation Plot
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Series d$resid_q
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Partial Autocorrelation Plot
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Lag
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tial A
CF
Series d$resid_q
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lag_09 lag_10 lag_11 lag_12
lag_05 lag_06 lag_07 lag_08
lag_01 lag_02 lag_03 lag_04
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Auto regressive errors
#### Call:## lm(formula = resid_q ~ lag_12, data = d_ar)#### Residuals:## Min 1Q Median 3Q Max## -12286.5 -1380.5 73.4 1505.2 7188.1#### Coefficients:## Estimate Std. Error t value Pr(>|t|)## (Intercept) 83.65080 201.58416 0.415 0.679## lag_12 0.89024 0.04045 22.006
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Residual residuals
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Residual residuals - acf
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Series l_ar$residuals
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lag_09 lag_10 lag_11 lag_12
lag_05 lag_06 lag_07 lag_08
lag_01 lag_02 lag_03 lag_04
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Writing down the model?
So, is our EDA suggesting that we then fit the following model?
sales(t) = β0 + β1 t+ β2 t2 + β3 sales(t − 12) + ϵt. . .
the implied model is,
sales(t) = β0 + β1 t+ β2 t2 + wt
where
wt = δ wt−12 + ϵt
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Discrete Time Series