Time series analysis
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What is a time series?
a set of observations collected over time a collection of random variables indexed by time
What is a time series process?
a collection of random variables indexed by time (X1, X2, …, XN) and a realization/instantiation of it (x1, x2, …, xN)
What does it mean to model a time series?
To specify and fit a model for a time-indexed process {Xt} that captures systematic structure (e. g. trend and seasonality) and time dependence (e. g. autocorrelation) in order to explain the series and/or forecast future values.
What are referred to as bull and bear markets in TSA?
periods of increments & decrements
What is a superposition in TSA?
the addition of 2 or more time series
How many cyclic patterns can a time series comprise?
more than 1
What is white noise in statistics?
a set of uncorrelated observations with 0 mean and unit variance no clear pattern TS after detrending and removing seasonality
What are general assumptions for white noise in stats?
• E = 0 • Var is constant • Cov(Xt, Xs) = 0, ∀ t ≠ s
According to Fan and Yao (2003), when does a TS {Xt} (t ϵ Z) have weak stationarity?
• The mean is constant. • The variance is finite. • Cov between 2 different random variables of the process only depends on the time lag between them. 1) E(Xt) = μ (constant) 2) E(Xt2) < ∞ for each t 3) Cov(Xt, Xt+k) = γ(k) (depends only on lag k)
How do we write the classical time series decomposition of Xt into a mean function and a residual term?
Xt = μt + Yt (t = 1, …, N) where Yt ~ WN(0, σ²) ► E[Yt] = 0 Var(Yt) = σ² Cov(Yt, Yt+k) = 0 for k ≠ 0
How can we write the mean function μt when a time series has an additive trend and seasonal component?
μt = Tt + St St is the seasonal component at t 𝑇t is the trend μt = E(Yt) = Tt + St
What is the abstract formula for correlation ρ between 2 random variables X and Y?
ρ(X, Y) = E[(X - E[X])(Y - E[Y])] / √(Var(X)*Var(Y)) -1 ≤ ρ(X,Y) ≤ 1 defined when Var(X) > 0, Var(Y) > 0 Correlation hits ±1 only when one variable is an exact affine (line-plus-intercept: Y = aX + b) function of the other, except possibly on a probability-zero set.
What is the formula for covariance γ between X and Y?
γ(X,Y) = E[(X - E(X))*(Y - E(Y))]
Most generally, what positive correlation is considered high?
anything above 0.8
What is spurious correlation?
cases where unrelated variables seem to walk together
How is the first difference in a time series denoted symbolically?
Δxt = xt - xt - 1
What is the k-lagged time series of x1, …, xN?
x1 - k, …, xN - k
What are common alternative names for autocorrelation (correlation of a series with its lagged values)?
serial correlation, lag(-k) correlation, self-correlation Related term: autocovariance = the unnormalized version of autocorrelation
How do we define autocovariance symbolically between 2 elements of a time series from time point t and s?
γ (t, s) = E[(Xt - E(Xt))*(Xs - E(Xs))] for all t, s ε Z
Why does the autocovariance of a weakly stationary time series Xt depend only on the lag, not on time position?
Weak stationarity requires that E[Xt] = μ and Cov(Xt, Xt+h) are both independent of t. Therefore autocovariance is a function of h alone: γ(h) = E[(Xt − μ)(Xt+h − μ)] Since γ(h) = γ(−h), we can equivalently write γ(k) where k = |h| ≥ 0.
Please give the sample autocovariance function defining ŷ(k) in the context of time series analysis.
ŷ(k) = 1/N * Σt = 1N - k ((xt - μ)*(xt + k - μ)) for k = 0, 1, …, N - 1
How do we express symbolically that the sample autocovariance function is even in the lag?
γHAT(k) = γHAT(-k) for all k ∈ ℤ
Please give the sample autocorrelation function expressed using the autocovariance?
ρHAT(k) = γHAT(k)/γHAT(0)
How can we transform a time series to have E(Xt) = 0?
We subtract the mean.