Chapter 3 - Basic concepts - Part B

Question 1

AR(p) is given by :

Answer 1

yt = φ1yt−1 + φ2yt−2 + ··· + φpyt−p + εt, (1) where φ1,φ2,…,φp are unknown parameters, and εt is a standard
white noise process.

Question 2

MA(q) is given by

Answer 2

yt = εt + θ1εt−1 + … + θqεt−q. where θ1,θ2,…,θq are unknown parameters.

Question 3

ARMAX( p, q) is given by

Answer 3

yt = α+φ1yt−1 +···+φpyt−p+ εt + θ1εt−1 + … + θqεt−q
(+β1x1,t + β2x2,t + · · · + βkxk,t),
where xi,t, i = 1, . . . , k denote (exogenous) regressors.

Question 4

How to implement ARMAX models in practice?

Question 5

We estimate AR(p) with

Answer 5

OLS

Question 6

We estimate MA(p) with

Answer 6

nonlinear least squares or maximum likelihood.

Question 7

Remember, b_ols =

Answer 7

see formula slide 6.

Question 8

Derive the unbiased and the variance of b_OLS and give the assumptions that were made.

Answer 8

E[b_OLS] = B
the diagonal elements of
V[bOLS] = (X′X)−1X′ΩX(X′X)−1

Question 9

Standard errond of b_OLS are the square roots of

Answer 9

the diagonal elements of
V[bOLS] = (X′X)−1X′ΩX(X′X)−1

Question 10

If εt is homoskedastic (E[ε2t ] = σ2 for all t) and uncorrelated (E[εtεs] = 0 for all t and s), Ω = σ2I, such that

Answer 10

Then V [bOLS] = σ2(X′X)−1

Question 11

If εt is heteroskedastic (E[ε2t ] = σt2) and uncorrelated, we have Ω = diag(σ1^2,σ2^2,…,σt^2), such that

Answer 11

see formula slide 9.

Question 12

What is a difference between the regressors of the classical linear regression model and an AR(p) model

Answer 12

The regressors of AR(p) are stochastic opposed to fixed.

Question 13

What does it imply when does it imply that regressors are stochastic?

Answer 13

It means that exact finite-sample results which hold in the ‘classical’ linear regression model yt = x′tβ + εt with fixed xt’s do not hold in the time series context.
⇒ Asymptotic results continue to hold, however.
For example, the OLS estimator of φ1 in the AR(1) model yt = φ1yt−1 + εt, is not unbiased but remains consistent.

Question 14

suppose a series yt is generated from the AR(1) model, yt = φ1yt−1 + εt,
What is the distribution of the OLS estimator of φ1?

Answer 14

See derivation.
√T(^φ1−φ1)∼N(0,σ^2γ0^-1 ) (ASYMPTOTIC relation!!)
Asymptotic approximation.

Question 15

The exact small-sample distribution can be obtained by means of …..:

Answer 15

a. Monte-carlo simulation
b. see slide 15.

Question 16

Two ways of model selection

Answer 16

• model selection criteria based on in-sample fit
• out-of-sample forecasting

Question 17

Give two model selection criteria

Answer 17

AIC : AIC(k) = T log σˆ2 + 2k,
SIC : SIC(k) = T logσˆ2 +klogT
=> Select ARMA orders p and q that minimize AIC(k) or SIC(k)

Question 18

Misspecification tests and diagnostic measures

Answer 18

Many tests aim to test whether the residuals of the ARMA model statisfy the white noise properties E[ε2t ]= σ2, and E[εtεs] = 0,

Question 19

Misspecification tests : test of no residual autocorrelation

Answer 19

see slide 18 and 19

Question 20

Misspecification tests : test for homoskedasticity

Answer 20

often based on autocorrelations of squared residuals.
⇒ If rejected, standard errors of parameters should be adjusted or heteroskedasticity should be modelled explicitly.

Question 21

Misspecification tests : test of normality

Answer 21

see slide 20

Question 22

3 different type of forecasting

Answer 22

1. A point forecast of y_T+h
2. An interval forecast : (^L_T+h|T,^U_T+h|T)
3. A density forecast : f(yT+h|YT)

Question 23

optimal h-step ahead point forecast depends on a …

Answer 23

loss function
We should use the point forecast ^y_(T+h|T)that minimize the expected value of the loss function.
The form of the loss function depends on the variable that we are forecasting.

Question 24

In many cases, the relevant loss function is diffucult to specify. Thus, we use the forecast error A . Most often, we assume that the forecast user has a B , that is C. Minimising C, or the D, we find the optimal point forecast is the E of E, that is F.

Answer 24

a. e_(T+h|T) = y_(T+h|T) − ^y_(T+h|T)
b. squared loss function.
c. Loss_(T+h|T) = e_(T+h|T)^2.
d. Mean squared prediction error [MSPE].
e. Conditional mean of y_(T+h|T)
f. ^y_(T+h|T) = E[ y_(T+h|T)|Y_T ]

Question 25

Derive the optimal point forecast of y_T+1 for an AR(1) model,
Given :
- E[εt|Yt−1] = 0
- E[εt^2 |Yt−1] = σ^2

Answer 25

see slide 25.
^y_(T+1|T) = E[Y_T+1 |Y_T ]
= E[φ1yT + ε_(T +1)|Y_T ]
= φ1yT

Question 26

Derive the relationship between e_T+1|T and ε_T+1 in the one-step ahead point forecast in the AR(1) model.

What conclusions can you draw?

Answer 26

• e_T+1|T = y_(T+1) − ^y_(T+1|T) = y_T+1 −φ1y_T = ε_(T+1)
• Hence, the variance of the forecast error V[e_(T+1|T)] is equal to σ^2, which is the variance of εt and also the conditional variance V[y_(T+1)|Y_T].

Question 27

what can you see about two steps ahead in the point forecast of a AR(1) model?
and 3 steps ahead?

Answer 27

see slide 27-28.

Question 28

Generalise the point forecasts of the AR(1) model for h-steps ahead.

Answer 28

See slide 29.

Question 29

Consider the AR(1) model with intercept and with E[ε_t|Y_t−1] = 0 and E[ε_t^2 |Y_t−1] = σ^2.

What are the optimal point forecast of y_T+1, y_T+2, and for 3 steps ahead?

What is the general h-steps ahead point forecast?

what happens when h converges to infinity (assuming |φ1| < 1) ?

Answer 29

see slide 30.

Question 30

The converge of point forecasts (with intercept) when the forecast horizon increases is even easier seen by rewriting the AR(1) model as A.
Using this representation, derive the conclusion that when h increase AR(1) model with intercept converges to the B.

Answer 30

a. yt −μ = φ1(yt−1 −μ)+εt
b. unconditional mean.
see slide 32 for the derivation.

Question 31

What is the effect of estimation uncertainty?

Answer 31

In practice, the true value(s) of the model parameter(s) are unknown. Instead, we have to use estimated parameters.
^y_(T+1|T) = ^φ1y_T.
Hence, e_(T+1|T) = y_(T+1) − ^y_ (T+1|T)
=φ1y_T + ε_T+1 − ^φ1y_T
= ε_(T+1) + ( φ1 − ^φ1 ) y_T .

Question 32

What is the effect of estimation uncertainty in point forecasts?

Answer 32

Mostly affects the variance of the forecast error V[e__(T+1|T)].
see slide 35.

Question 33

What are the effects of model misspecification in point forecasts ?

Answer 33

see slide 37.

Question 34

How to evaluate point forecasts?

Answer 34

Two ways, absolute or relative evaluation.
Absolute evaluation: what is the quality of forecasts form one specific model?
Relative evaluation: wha tis the quality of the forecasts form multiple competing models, relative to each other.

Question 35

What are the 3 desirable properties of the point forecasts ?

Answer 35

1. Unbiasedness : forecast errors have the zero mean E[e_(t+1|t)] = 0 => Straightforward to examine by testing the mean of the forecast errors differs significantly form 0.
2. Accuracy: MPSE should be as small as possible.
   Recall that the point forecast ^y_(t+1|t) (usually) is taken to be the one which minimizes
   E[e_(t+1|t)^2 ] = E[(y_(t+1) − ^y_(t+1|t) )^2]

Note that the MSPE can be decomposed as
… = variance + squared bias

3. Efficiency / optimality: it should not be possible to forecast the forecast error itself with any information available at time t. see slide 44.

Question 36

Evaluating point forecasts

Answer 36

see slide 42-43

Question 37

Comparing predictive accuracy

Answer 37

slide 45

Question 38

How to construct density forecasts?

Answer 38

slide 47

Question 39

How to construct interval forecasts?

Answer 39

slide 48