Stationary

Question 1

What is a stochastic process?

Answer 1

A random process that evolves over time e.g. x(t) = θx(t-1) + ε(t) (this is an AR process)

Question 2

Is the probability distribution of ε(t) dependent or independent of time?

Answer 2

It is independent of time

Question 3

How do we write and AR process in MA form?

Answer 3

By backward substitution we have: x(t-1) = θx(t-2) + ε(t-1) x(t-2) = θx(t-3) + ε(t-2)… we come to: x(t) = ε(t) + θε(t-1) + θ²ε(t-2) … -> ∑θ(i)ε(t-i)

Question 4

What assumptions do we make about this white noise?

Answer 4

• It is Gaussian
• its expected value is 0 with constant variance, following a normal distribution.
• The error terms are uncorrelated i.e. E(ε(t),ε(t-1))=0

Question 5

What is the necessary condition for an AR(1) process to be stationary?

Answer 5

Autocorrelations must approach zero exponentially - |θ| < 1 In order for |θ| < 1, L > 1 (the invertibility condition)

Question 6

How do we calculate the value of Φ(1) and Φ(2)?

Answer 6

Through expansion thing:

• Find the equation with the lag operators (i.e. 1-0.1L+1.2L²=0)
• Find the value for L(1) and L(2)
• Find Φ(1) and Φ(2) through the equation (1-Φ₁L)(1-Φ₂L)=0

Question 7

Is it possible to estimate the parameters of the process given that we only observe one realisation?

Answer 7

Yes, assuming the process is ergodic i.e. if the sample moments of a particular realisation approach the population moments of the process as the length of the realisation becomes infinite

Question 8

Is it possible to test for ergodicity?

Answer 8

No, so we really on the weaker property of stationarity

Question 9

What is a strictly stationary stochastic process?

Answer 9

A stochastic process whose properties are unaffected by a change of time origin i.e. x(1) at t(1) must have the same joint probability distribution as x(1+k) at t(1+k)

Question 10

What is a weak or covariance stationary stochastic process?

Answer 10

One where: E[x(t)] is constant across time V[(x(t)] is constant across time Cov[x(t),x(t-k)] and Corr[x(t),x(t-k)] depend only on the lag k but not time

Question 11

What’s the difference between Autocorrelation and Partial-Autocorrelation?

Answer 11

PAC uses the method of moments to compute, whereas AC uses OLS
With AC, we compute the AC between x(t)→x(t-1), then x(t)→x(t-2) directly, the PAC takes into account all the steps in between x(t)→x(t-k)

Question 12

Are MA(1) processes stationary?

Answer 12

Yes because the value variance does not vary over time, nor does covariance

Question 13

Why are PACs useful?

Answer 13

They are useful to capture the correlation between the series and its past value while allowing for the intermediate effects of lags

Question 14

What are the Yule-Walker equations?

Answer 14

ρ(1) = θ(1) + θ(2)ρ(1) 
ρ(2) = θ(1)ρ(1) + θ(2)

Question 15

What is the calculation for correlation? (ρ)?

Answer 15

ρ(1) = γ(1)/γ(0) ρ(2) = γ(2)/γ(0)
OR
ρ(1) = θ(1)/1-θ(2) ρ(2) = θ(1)²/(1-θ(2)) + θ(2)
where γ(k) is the covariance between x(t),x(t-1) and γ(0) is variance of x(t)

Question 16

What is the MA(1) equation?

Answer 16

x(t) = ε(t) + αε(t-1) It is a function of the error terms and their past values weighted by the value, α

Question 17

What is the expected value of x(t) in an MA(1) process?

Answer 17

0

Question 18

What is the variance (γ(0)) of the process?

Answer 18

σ²(1+α²)

Question 19

What is the covariance (γ(1)) = E[x(t)x(t-1)] of the process?

Answer 19

ασ²

Question 20

What is the covariance (γ(k)) = E[x(t)x(t-k)] of the process?

Answer 20

0, where k≥2

Question 21

Is the MA process weakly dependent?

Answer 21

Yes, because the autocorrelation moves quickly to 0

Question 22

What is the Stationarity condition of an ARMA(p,q) process?

Answer 22

|θ|<1 where: θ(L) = (1-Φ₁L)(1-Φ₂L)…=0

Question 23

What is the invertibility condition of an ARMA(p,q) process?

Answer 23

where: α(L) = (1-h₁L)(1-h₂L)…=0

|h|<1, L>1

Question 24

What are the steps to the Box-Jenkins method to fit an ARIMA process to a model?

Answer 24

1. Identification (examine the correlogram) 2. Estimation (Fit an AR(1) process to the data) 3. Diagnostic Checking (examine correlogram of the residuals (again))

Question 25

What is the Box-Jenkins philosophy? (i.e. how to choose which model is the best fit if a few work?)

Answer 25

Select the most parsimonious one - the one with the fewest parameters i.e. an ARMA(1,1) is preferred to an ARMA(3,0)

Question 26

How do we identify a unit root process?

Answer 26

The AC will decrease linearly and very slowly

Question 27

Is Bayesian Information Criterion (BIC) stricter than Akaike Information Criteria (AIC)?

Answer 27

Yes, because it leads to lower values of p and q being chosen in the ARMA(p,q) model

Question 28

How do we choose the best AIC and BIC?

Answer 28

When estimating a range of ARMA(p,q) models we look for the lowest values in AIC and BIC

Question 29

How do we assess the performance of a forecasting model? (2 methods)

Answer 29

Root Mean Squared Errors

Theil’s U statistic

Question 30

What is the formula for RMSE?

Answer 30

√(1/k)∑(x-x^) This compares different models (note: the √ is over everything) x is the observed value and x^ is the forecasted value

Question 31

What is the formula for Theil’s U statistic?

Answer 31

RMSE/(√(1/k)∑x²)+(√(1/k)∑x^²) if U = 0, model predicts the data perfectly U = 1, negative/opposite relation between actual and predicted values

Question 32

How do static and dynamic forecasts differ depending on how far ahead we are forecasting?

Answer 32

If it is only 1 period ahead, static = dynamic If we are forecasting two or more ahead: -Static → substitute actual values into the forecasting equation → information set is lagged only one period in the past → x is used to forecast x^ -Dynamic → substitute forecast values into the forecast equation → information set is fixed at date T → x is used to forecast x^(1), x^(1) is used for x^(2)… -Static forecasts have lower forecast error variance i.e. lower s.e. bands -s.e. = RMSE remains constant for static but increases for dynamic