NonStationarity Flashcards
What is a random walk process?
A process where the current value of x is equal to its value in the last period plus a random disturbance value, ε
What is the formula for a random walk process?
xt = xt-1+εt (note: this is an MA process where θ=1)
Does a disturbance far in the past have the same influence on xt as a more recent disturbance?
Yes
Is the Variance constant?
No, it is a value σ² which does not converge/ is infinite. Hence it is non-stationary
What is an integrated process of order 1?
A process with one unit root that can be made stationary by differentiating it one time
How do we difference a process?
We form the equation with ‘L’ and factorise this
We note that (1-L)yt = yt-yt-1 = Δyt
i.e. xt=1.3xt-1-0.3xt-2+εt
(1-1.3L+0.3L²)xt=εt
(1-L)(1-0.3L)xt=εt
Δxt=0.3Δxt-1+εt
How do we represent the second difference?
(1-L)²yt=Δyt-Δyt-1=Δ²yt
What does stationarity depend on in a linear process with normally distributed disturbances? (2)
Mean and Variance
What is a Deterministic Trend Process?
A process where our series is simply a function of a constant, time and a disturbance i.e. x(t) = α + βt +ε(t) This is nonstationary due to the time variable It is, however, trend stationary because it evolves around a common trend
How do we make a Deterministic trend process stationary?
If we know the trend, we can make a new variable, z(t), which is the difference between the old series and the trend. This will remove the trend (detrending)
What is the value of z(t)?
z(t) = x(t) - βt
What is a Random Walk with Drift?
It is a random walk with a constant that is different from 0 x(t) = α + x(t-1) + ε(t) E[x(t)] = αt (i.e. not constant over time) V[x(t)] is not defined
How do we make a random walk with drift stationary?
Through differencing Δx(t) = α + ε(t)
What is the formula for a Random walk with drift and deterministic trend?
x(t) = α + βt +x(t-1) + ε(t)
What is the expected value of x(t) and the Variance of x(t) in this process (Random Walk with Drift and Deterministic Trend)?
E[x(t)] = ƒ(t) = V[x(t)] so time is important for both
How do we make this series stationary (Random Walk with Drift and Deterministic Trend)?
- Difference (Δx(t) = α + βt + ε(t))
- Detrending (z(t) = Δx(t) - α - βt, z(t) = ε(t)
How can we use the AC and PAC to determine if a process is stationary or nonstationary?
A nonstationary process will have declining ACs but in a linear rather than and exponential pattern
How do we determine a random walk process through correlograms?
If there are no ACs or PACs, it is a random walk process
How do we use correlograms to identify unit root?
ACs will be decreasing linearly and very slowly
The PAC will be ~1 and then non existent
What is the null hypothesis of Dickey-Fuller unit root test?
H₀: The process is unit root, when ρ=1
H₁: The process is stationary, when ρ<1
Why do we not care about the case where ρ>1 or ρ<0?
ρ<0 are rare explosive ones, ρ>1, aren’t usually considered
How do we carry out the Dickey-Fuller test?
Subtract x(t-1) from both sides of x(t)=α+ρx(t-1)+ε(t)
Define θ=ρ-1
Hence, Δx(t)=α+θx(t-1)+ε(t)
What is the null hypothesis of the DF with θ?
H₀:θ=0 (unit root or I(1))
H₁:θ<0 (stationary or I(0))
What is the problem with the DF test?
The t-statistic=θ^-θ/se(θ^) does not follow the usual t distribution (or approximates the standard normal as the sample size increases)
When do we reject the null of the DF test?
reject H₀:θ=0 if tDF
Why do we augment the Dickey-Fuller test?
We augment to control for the presence of serial correlation in the test equation (in order to ensure the standard error of the coefficient estimate for xt-1 is not biased)
How do we augment the Dickey-Fuller test?
We use the lagged changes ∆x(t-h), h=1,2,…,p
e.g. ∆x(t)=α+θx(t-1)+γ∆x(t-1)+ε(t)
How do we perform the ADF test?
Regress ∆x(t) on x(t-1),∆x(t-1),…,∆x(t-p) (include ∆x(t-h) needed to avoid serial correlation)
-only include lags that are significant in the estimations of this test
Carry out the t-test on θ^ as before (critical values and rejection rule same as before)
How do we perform an ADF test with drift and deterministic trend?
∆x(t)=α+βt+θx(t-1)+∑γ∆x(t-i)+ε(t)
As before,
H₀:θ=0 (unit root or I(1)) - reject H₀:θ=0 if t < cDFtrend
H₁:θ<0 (trend stationary or I(0))
Note: with a trend the critical values changes
What is the main issue with an ADF test?
It has a lack of power, i.e. it is unable to reject a false null hypothesis
This is because the alternative hypothesis includes a wide range of values (some very close to the null)
What alternative tests are there?
KPSS, Phillips-Perron, Modified ADF test
What is unique about the KPSS test?
It is the only test listed in which the null hypothesis is stationarity
H₀: Stationarity
H₁: Unit root
How does the KPSS test work?
- Regress x(t) on a constant (and trend, if it exists)
- save residuals ε^(t) and compute the partial sums S(t)=∑ε^(s) for all values of t
- Test statistic KPSS=(1/T²)∑[S(t)²/σ^²]
- If KPSS>c reject H₀
Why is KPSS a stronger test?
It helps to mitigate the ADF power problem in small samples
How does the Phillips-Perron test work?
∆x(t)=α+βt+θx(t-1)+ε(t)
- Similar to DF
- The bias in the standard errors is adjusted using non-parametric methods
- They modify the test statistics to account for any autocorrelation effects
- The critical values come from DF
What is the decision rule of the Phillips-Perron test?
Decision rule: reject the null if test statistic lower than critical value
What is the drawback of the Phillips-Perron test?
It tends to perform less well than ADF in finite/small samples
How does the Modified ADF test (DF-GLS) work?
View lecture slide 33 for equation GLS transformation prior to the estimation of the Dickey-Fuller regression
Employing quasi-differences (x(t)-πx(t-1)) to remove the drift and trend from the series and obtain the detrended series xdt=xt-φ^(drift)-φ^(trend)t
The φ’s come from the estimation of the x(t)-πx(t-1) on 1-π and t-π(t-1)
What is the decision rule of the Modified ADF test?
Reject the null if test statistic is lower than the critical value
Do these tests work for multiple unit roots?
No, so we need to difference to I(0) to make it stationary
How does forecasting with a random walk with drift [x(t)=α+x(t-1)+ε(t)] differ from forecasting with a stationary series?
E[x(T+k)|I(T)] = x(T)+kα ←kα
V[(x(T+k)|I(T)] = kσ² ←k
Variance increases the further into the future we want to predict
Our confidence declines
How does forecasting with a trend stationary process [x(t)=α+βt+ε(t)] differ from forecasting with a stationary series?
E[x(T+k)|I(T)] = [x(t)-ε(t)]+βk ←-ε(t)+βk
V[(x(T+k)|I(T)] = σ² ←no change
What is seasonality?
A phenomenon whereby something is more likely to occur in certain quarters than others (i.e. ice cream sales are probably higher in q3 than any other)
How do we deal with seasonality?
1.Try to check for this pattern while looking at the differences
How do we fix a model with pure seasonality?
We can estimate the model with only the 4th lag i.e. x(t)=θ₄xt-4+εt
What is the issue with this method?
View pic
How do we fix a model with a mixture of seasonal and non-seasonal?
We can estimate a model which is dependent on both the 1st and 4th lag i.e. xt = θ₁xt-1 + θ₄xt-4 + εt -this allows for autoregressive effect plus seasonal effect (4th lag)
What alternative approach can we use?
Introducing a multiplicative seasonal factor
- this implies an interaction between seasonal and non-seasonal parts
