forecasting and time series 4-6

stationary

the general linear process is (stationary/not stationary)

nonstationary

With a _ series, the ACF typically does not die off quickly as the lag increases.

larger

for _ sample sizes, the approximate sampling distribution of rk cane found, when the data come from ARMA-type models

no die when go back in time

for the IMA(1,1) process the weights on the et's

AR(p)

for this model the autocorrelations don't cut off at a certain lag; they die off gradually toward zero

jagged

for φ near −1, the overall graph of the process will appear _

AIC

=−2logL+2k, where L here is the maximized likelihood function and k = p + q + 1 for a model with an intercept term and k = p + q for a model without an intercept.

zero weaker

AR(1) process, Since |φ| < 1, the autocorrelation gets closer to _ or _ as the number of lags increases.

recent

AR(1), the value of the process is associated with very _ values

ARIMA

Auto Regressive Integrated Moving Average

stationary invertible

For ARMA(p,q) models, we restrict attention to those models which are both _ and _

smooth

For φ near 1, the overall graph of the process will appear _

integrated moving average

If the ARIMA process has no autoregressive terms

strongly positive

IMA (1,1) as time goes on correlation between values of the process will be _ _ for small lags

larger

IMA (1,1) as time goess on var(Yt) gets

autoregressive integrated process

If the ARIMA process has no moving average terms

yule-walker equations

If the general AR process is stationary, these can be formed

normally distributed

If the model of AR(p) is correct, then the sample partial autocorrelations for lags greater than p are approximately _ _ with means 0 and variances 1/n.

PACF

If we use an AR model for these data, we should examine the _ to determine the order of the AR model.

conditional bivariate

If {Yt} is a normally distributed time series, the PACF can be defined as the correlation coefficient of a _ _ normal distribution: φkk = corr(Yt,Yt−k|Yt−1,...,Yt−k+1)

general linear process

Let {Yt} be our observed time series and let {et} be a white noise process (consisting of iid zero-mean r.v.'s). {Yt} is a general linear process if it can be represented by: Yt = et + ψ1et−1 + ψ2et−2 + · · · where et , et−1, . . . are white noise.

nonunique

MA(1) process is _, we get the same autocorrelation function if we replace θ by 1/θ.

1. choosing appropriate values for p,d,q 2. estimating the parameters 3. checking model adequacy

Model specification for ARIMA(p,d,q) models involves:

noninvertible

Overdifferencing will create a _ model, which leads to problems with interpretability and parameter estimation.

do not have evidence against

Sample PACF, for any lag k > p, if the sample partial autocorrelation φˆkk is within 2 standard errors of zero (between −2/√n and 2/√n), then this indicates that we _____________ the AR(p) model.

invertible

So the MA(1) model is _ if and only if |θ| < 1

exceed

The AR(p) process is stationary if and only if the solutions of the AR characterstic equation _ 1 in absolute value

p q

The EACF table for an ARMA(p, q) process should theoretically have a triangular pattern of zeroes with the top-left zero occurring in the _-th row and _-th column (with the row and column labels both starting from 0).

correlation

The PACF at lag k is denoted φkk and is defined as the _ between Yt and Yt−k after removing the effect of the variables in between: Yt−1, . . . , Yt−k+1.

The null hypothesis is that the series is nonstationary, but can be made stationary by differencing.

The _ _ of the dickey-fuller unit root test is that the series is nonstationary, but can be made stationary by differencing.

ARIMA(p,d,q)

The _ class of models as a broad class can describe many real time series

Dickey-Fuller Unit Root

The __________ test is a formal hypothesis test for whether the time series is "difference nonstationary."

zero

The autocorrelations ρk are _ for k > q and are quite flexible, depending on θ1, θ2, . . . , θq, for earlier lags when k ≤ q.

should not

The fact that the sample autocorrelations do not cut off after a certain lag tells us that we _______ use an MA model for this time series.

theta1 theta2

The formulas for the lag-k autocorrelation, ρk, and the variance γ0 = var(Yt) of an AR(2) process are complicated and depend on _ and _

invertible

We can solve the nonuniqueness problem of MA processes by restricting attention only to _ MA models.

ARIMA

We have seen that many real time series exhibit nonstationary behavior. For these _ models are better

exponential growth

When AR with |φ| > 1, we get an _ _ model in which the weights on past disturbance terms blow up (rather than dying out) as we go further into the past.

autoregressive moving average process of order p and q

a time series that has both autoregressive and moving average components

sample autocorrelation function

a useful tool to check whether the lag correlations that we see in a data set match what we would expect under a specific model

rk

are estimates of patterns of the ρk for ARMA models

simple sampling distribution

because of its form and the fact that it is a function of possibly correlated values, rk does not have a _ _ _

positive

box-cox transformation assumes the data values are all

arma(p,q)

can be found numerically solving a series of equations that depend on φ1, . . . , φp or θ1,...,θq.

partial autocorrelation function

can be used to determine the order p of an AR(p) model

first differences

d = 1

0

general linear process expected value

present and past white noise terms

general linear process is a weighted linear combination of

linearly

if Yt is changing exponentially, the logged series will change

constant

if box-cox transformation data values are not all positive add a _ to make them positive

white noise

if the original data follow a random walk process, then taking first differences would produce a stationary _ _ model.

zero

in the AR(p) process, φkk = _ for all k > p.

MA(q)

in this model we know the autocorrelations should be zero for lags beyond q

IMA(d,q)

integrated moving average

extended autocorrelation function

is one method proposed to assess the orders of a ARMA(p, q) model.

Box Jenkins Method

process of iteratively proposing, checking, adjusting, and re-checking the model for fitting time series

autoregressive processes

process that takes the form of a regression of Yt on itself, or more accurately on past values of the process

non stationary

process with a mean function that varies over time

disturbance term

so the value of the process at time t is a linear combination of past values of the process plus some independent _ _

moving average

special case of the general linear process

model adequacy

strategy for checking _ _ is to see whether the rk's fall within 2 standard error of their expected values (the pk's)

stationary

taking the natural log and then taking first difference should produce a series that is

MA(q)

the PACF for a _ process behaves similarly as the ACF for an AR process of the same order

shapes

the autocorrelation function can have a wide variety of _, depending on the values of φ1 and φ2

increases

the autocorrelation ρk dies out toward 0 as the lag k _

decays increases

the partial autocorrelation of an MA(1) process never equals zero exactly, but it _ to zero quickly as k _.

exponentially decaying

we often assume that weights in the general linear process are

reciprocal transformation

when box-cox lambda = -1

square root

when box-cox lambda = 1/2

regularity condition on the coefficients

when the number of terms in the general linear process is actually infinite we need some

BIC

−2logL+klog(n),