4th year Flashcards
(16 cards)
define stationarity
A time series {yt} is said to be stationary if
(i) the mean of yt does not depend on t and
(ii) the covariance between yt and yt+s is a function of s only.
why is stationarity a useful assumption
for the estimation of the covariance between yt and yt+s.
This assumption is need to estimate the common mean by averaging y1, . . . , yn,
and the common covariance between yt and yt+s by averaging over all such pairs
of data.
system equation and observation equation for the Kalman filter
The Kalman filter has
System equation: xt = F x_t−1+v_t, t = 1, 2, . . .,
Observation equation: yt = Gxt + wt, t = 1, 2, . . ..
Statistical properties of error terms in Kalman filter
The assumptions on the error terms are
{vt} are i.i.d. multivariate normal with mean vector 0 and cov matrix Q;
{wt} are i.i.d. multivariate normal with mean vector 0 and cov matrix R;
{wt} are independent of {vt}.
What problem does the Kalman filter solve
The Kalman filter solves the problem of finding [Bookwork]
xˆt|t
= E[xt|y1, . . . , yt] and Pt|t = var{xt − xˆt|t}.
how does Kalman filter solve the problem
It solves the problem recursively by finding
xˆt|t−1= E[xt|y1, . . . , yt−1] and Pt|t−1 = var{xt − xˆt|t−1} first in each iteration.
What other models (than the ARMA type) can be fitted using the Kalman filter?
Example here is CAPM with time-varying beta. Other examples include random
walk plus noise model and local linear trend model.
Dynamic linear model: yt = at + btxt + εt, εt ∼ iid N(0, σ2), at = at−1 + vt , vt ∼ iid N(0, σ2a), bt = bt−1 + wt, wt ∼iid N(0, σ2b), {εt}, {vt} and {wt} are independent.
Define σ^2-t for a GARCH model
Definition. σ2t = Var{yt|yt−1,yt−2, . . .}.
Assumptions on {ε_t} for GARCH
It is assumed that {εt} are iid with mean zero and variance one, and that εt
is independent of yt−1, yt−2, . . ..
Autocorrelation
ρ(s) = corr{yt, yt+s}, s = 0, ±1, ±2, . . . .
Partial autocorrelation
The partial autocorrelation function is φkk, k = 1, 2, . . ., where φkk is the coeffi-
cient of yt−k in the best linear predictor of yt
in terms of yt−1, yt−2,. . . , yt−k
Describe Kalman smoothing
Kalman smoothing works backwards from t = n to t = 1 to give
xˆt|n= E[xt|y1, . . . , yn] and Pt|n = var{xt − xˆt|n}.
Define σ^_t in GARCH model
In the GARCH(1,1) model, σ
2tis the conditional variance of yt given yt−1, yt−2,. . . (infinite past).
Why is the Kalman filter used in maximum likelihood estimation of ARMA models?
The Kalman filter is used in ML estimation of ARMA models because it gives
as by-products et = yt − yˆt|t−1 for t = 1, 2,…,n which are uncorrelated and have
the same log-likelihood as y1,…, yn.
What is the reduced log-likelihood of the Kalman filter
The reduced log-likelihood is the log-likelihood of y1,…, yn given the AR and
MA parameters, with σ2
replaced by its optimal solution in terms of φe
and θe.
How is the reduced log likelihood maximised using the Kalman filter?
To maximise a function, it must be calculatable given the variable values. The
Kalman filter makes the (reduced) log-likehood function calculatable by supplying et’s and τt’s based on the parameter values.
