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1

Time series

A set of observations made at ordered time-points.

2

The index set

Time points at which the process is defined

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State space

set of values that the random variables Xt may take

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Aims of time series

1. Draw inferences from time series 2. Examining the process generating the data 3. Forecasting

5

What is White Noise

1. E(εt)=02. Cov(εt,εs) = σε iff t=0 else 0

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Gaussian Stochastic process

All of its marginal distributions (any part of distribution) are Gaussian

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What is Gaussian White Noise

If WN is also Gaussian

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Are white noise shocks independent

White Noise shocks are only uncorrelated but not independent. Can have some dependents between absolute/squared shocks

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Why is Gaussian WN is independent

Normally distributed random variables are uncorrelated if and only if they are independent

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What is Random Walk

Xt=Xt-1+εt

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Random walk with drift

Xt=a0+Xt-1+εt

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Weak Stationarity

1. Xt has finite moments of second order for all t in Z2. E(Xt) = E(Xs) for all t,s in R3. Cov(Xt, Xs) = Cov (Xt+h,Xs+h) for all h in NIn Words First/Second order process variables don't depend on time

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Strong stationarity

Joint distribution doesn't depend on time For any consecutive m (from Z) and a lag h (from N) Xt1,...,Xtm and Xt1+h,...Xtm+h are identical

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ACVF (AutoCoVariance Function)

γ(h)=Cov(Xt,Xt+h)

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ACF (AutoCorrelation Function)

ρ(h)=Corr(Xt,Xt+h)

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Properties of ACF and ACVF

1. Positive semidefinite 2. Symmetric

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What does it mean if an ACF or ACVF matrix is not positive semidefinite

Process is not stationary

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What do ACF and ACVF measure

Degree of dependence among the values of a time series at different times

19

What is the general approach to Time Series Modelling

1. Plot the series and examine the main features of the graph, checking in particular whether there is: a) trend b)seasonal component c)any apparent sharp changes in behaviour d)any outlying observations2. Remove the trend and seasonal components to get stationary residuals 3. Choose model to fit the residuals 4. Forecasting will be achieved by forecasting the residuals and the inverting any transformations to arrive at forecasts of the original series

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Classical decomposition model

Xt=mt+st+Ytmt = trend functionst = seasonal componentYt=zero-mean random noise component

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What are deterministic component/s (signal) and what are stochastic component/s in the Classical decomposition model

mt and st are signalYt is noise

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What should do if Var increases

Apply preliminary transformations e.g. Log

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If models with trend, but no seasonality (Xt = mt + Yt) how estimate trend

Method 1: Trend estimation:a)Nonparametric: 1. Moving Average 2. Exponential smoothing b)Model based:Fitting a polynomial trendMethod 2: Trend elimination by differencing

24

Constant Mean Model (CMM)

Xt = m + Yt where m is a constant. Big problem is that assigned weights are equal

25

What is the usual trade of between choosing large or small q

If mall - Faster reactionIf large - smaller variability

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What method is used to estimate parameters in the polynomial trend model

Method of least squares

27

The lag-1 difference operator

∇Xt=Xt-Xt-1=(1-B)Xt

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Backward shift operator

BXt=Xt-1

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What is the problem of applying difference operator for a small sample size

Reduce sample size by 1 with each differencing

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∇^2Xt

Xt-2Xt-1+Xt-2

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Two methods to deal with Trend and Seasonality

Method 1: Trend and seasonality estimation a)Estimate the trend using a centered simple moving average b)Estimate the seasonal component c)(Optimal) Reestimate the trend of the deseasonalised data d)Estimate the noise using the (re)estimated trend and the estimated seasonal component. Method 2: Differencing

32

What if seasonal effect d is odd

q=(d-1)/2

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What if seasonal effect d is even

q=d/2 but assign 0.5 weight to the first and last Xt

34

∇dXt

Xt-Xt-в

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Is ∇d == ∇^d

No

36

What is the problem if h is close to T in the sample ACF/ACVF

Estimator is not very reliable

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Rule of thumb for T and h in (sample ACF/ACVF)

T>50 and h

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For Large T the sample ACF of an iid sequence with finite variance are approximately

~N(0,1/T)

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CI for ACF

estimate +- 1.96/sqrt(T)

40

H:0 and H:1 for Portmanteau Test

H:0 ρ(1)=ρ(2)=...=ρ(m)=0H:1 at least 1 ρ is not equal to 0

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How m is chosen for Portmanteau Test

Ad hoc

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Power of a test

P(alternative | alternative is true)

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How can test normality

Jarque-Bera testQ-Q plot

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How does Q-Q plot works

It plots an empirical quantiles against theoretical ones. In particular the values in the sample of data, in order from smallest to largest, are plotted against F-^-1 ((i-0.5)/T), i=1,2,...,T and А is the cumulative distribution function of the assumed distribution. If Assumed distribution is accurate then expect a 45 degree line through (0,0)

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For which processes do ACVF and ACF exist

Weakly Stationary

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For which processes do sample ACVF and ACF exist

Non weakly stationary

47

What does convergence in mean square allows

for infinite linear combinations of past values

48

Autocorrelation matrix

Rk:= (ρ(i-j))i,j=1,...,kSketch

49

α(h) measures what

Left over correlation between Xt and Xt-h after accounting for their correlation with Xt-1, Xt-2,...,Xt-h+2,Xt-h+1

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Markov process

E(Xt | Xt-1,Xt-2,...) = E(Xt | Xt-1)

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Martingale process

E(Xt | Xt-1,Xt-2,...) = Xt-1

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Example of Markov process

AR(1)

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Example of Martingale process

RW without drift

54

Define MA(1)

Xt=εt+θεt-1

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5 Properties of MA(1)

1. Xt is weakly stationary2. γ(h) = (1+θ)^2σ^2 h=0 θσ^2 |h|=1 0 otherwise3. ρ(h) = 1 h=0 θ/(1+θ^2) |h|=1 0 otherwise4. PACF5. |θ|<1 X is invertible with respect to ε εt=(Xt-μt)-θ(Xt-1-μ)+θ^2(Xt-2-μ)-+... |θ|>=1 X is not invertible with respect to ε |θ|>1 X is invertible with respect to WN z:=(Xt-μt)-1/θ(Xt-1-μ)+1/θ^2(Xt-2-μ)-+...

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What is decaying and what cuts of for MA

ACF - DecaysPACF - Cuts off

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How confidence interval for an empirical ACF is called

Bartlett

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Write an equation for MA(inf)

Xt=εt+θ1εt-1+θ2εt-2+...

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What is the other name for MA(inf)

General linear process

60

Define AR(1)

Xt=φXt-1+εt

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How stationarity property for MA is different from AR

In the AR it is explicitly postulated in the definition while in the MA is it not presupposed

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What happens to AR process if |φ|>1/=1/<1

<1 is weakly stationary>1 Can find the new white noise sequence for which Xt is stationary=1 Non stationary as is Random walk

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If all roots of the corresponding auxiliary equation lie inside the unit circle then the process is

Stationary

64

MA(q) is invertible with respect to ε if and only if

Roots of the MA-polynomial lie outside of the unit circle

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How is this equation and its roots are called? ψ(x)=x^p-φ1x^p-1-φ2x^p-2-...-φ^p=0

Characteristic equation and characteristic roots

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What happens when the characteristic roots are complex numbers

Mixture of damping sine and cosine patterns

67

What is decaying and what cuts of for AR

ACF - Cuts off PACF - Decays

68

Define ARMA(1,1)

Xt=φXt-1+εt+θεt-1

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What are the restrictions on the φ and θ in the ARMA (1,1)

φ+θ≠0 and both φ and θ are real numbers

70

What is the restriction imposed on the roots of AR and MA parts of the ARMA process

There can't be any common roots for the AR and MA process

71

When does ARMA have an infinite AR/MA representation

If the AR operator φ is stationary (i.e. all the roots of φ(x)=0 lie outside the unit circle) the ARMA process given by φ(B)Xt=θ(B)εt is indeed stationary and can be represented as an MA(inf) process. If the MA operator θ is invertible (i.e. all the roots of θ(x)=0 lie outside the unit circle) the ARMA process φ(B)Xt=θ(B)εt has an infinite AR representation.

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For ARMA(p,q) what does ACF mimics when q=>p after lag q-p

AR(p)

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For ARMA(p,q) what does PACF mimics when p=>q after lag p-q

MA(q)

74

Which selection criteria is/are consistent and which asymptotically efficient

AIC,AICc are asymptotically efficientBIC consistent

75

How do we choose a model based on AIC/BIC

Choose the one which has the lowest value