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Econometrics > NonStationarity > Flashcards

NonStationarity Flashcards

1
Q

What is a random walk process?

A

A process where the current value of x is equal to its value in the last period plus a random disturbance value, ε

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2
Q

What is the formula for a random walk process?

A

xt = xt-1+εt (note: this is an MA process where θ=1)

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3
Q

Does a disturbance far in the past have the same influence on xt as a more recent disturbance?

A

Yes

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4
Q

Is the Variance constant?

A

No, it is a value σ² which does not converge/ is infinite. Hence it is non-stationary

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5
Q

What is an integrated process of order 1?

A

A process with one unit root that can be made stationary by differentiating it one time

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6
Q

How do we difference a process?

A

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

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7
Q

How do we represent the second difference?

A

(1-L)²yt=Δyt-Δyt-1=Δ²yt

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8
Q

What does stationarity depend on in a linear process with normally distributed disturbances? (2)

A

Mean and Variance

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9
Q

What is a Deterministic Trend Process?

A

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

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10
Q

How do we make a Deterministic trend process stationary?

A

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)

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11
Q

What is the value of z(t)?

A

z(t) = x(t) - βt

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12
Q

What is a Random Walk with Drift?

A

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

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13
Q

How do we make a random walk with drift stationary?

A

Through differencing Δx(t) = α + ε(t)

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14
Q

What is the formula for a Random walk with drift and deterministic trend?

A

x(t) = α + βt +x(t-1) + ε(t)

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15
Q

What is the expected value of x(t) and the Variance of x(t) in this process (Random Walk with Drift and Deterministic Trend)?

A

E[x(t)] = ƒ(t) = V[x(t)] so time is important for both

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16
Q

How do we make this series stationary (Random Walk with Drift and Deterministic Trend)?

A
  1. Difference (Δx(t) = α + βt + ε(t))
  2. Detrending (z(t) = Δx(t) - α - βt, z(t) = ε(t)
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17
Q

How can we use the AC and PAC to determine if a process is stationary or nonstationary?

A

A nonstationary process will have declining ACs but in a linear rather than and exponential pattern

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18
Q

How do we determine a random walk process through correlograms?

A

If there are no ACs or PACs, it is a random walk process

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19
Q

How do we use correlograms to identify unit root?

A

ACs will be decreasing linearly and very slowly

The PAC will be ~1 and then non existent

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20
Q

What is the null hypothesis of Dickey-Fuller unit root test?

A

H₀: The process is unit root, when ρ=1

H₁: The process is stationary, when ρ<1

21
Q

Why do we not care about the case where ρ>1 or ρ<0?

A

ρ<0 are rare explosive ones, ρ>1, aren’t usually considered

22
Q

How do we carry out the Dickey-Fuller test?

A

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)

23
Q

What is the null hypothesis of the DF with θ?

A

H₀:θ=0 (unit root or I(1))

H₁:θ<0 (stationary or I(0))

24
Q

What is the problem with the DF test?

A

The t-statistic=θ^-θ/se(θ^) does not follow the usual t distribution (or approximates the standard normal as the sample size increases)

25
Q

When do we reject the null of the DF test?

A

reject H₀:θ=0 if tDF

26
Q

Why do we augment the Dickey-Fuller test?

A

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)

27
Q

How do we augment the Dickey-Fuller test?

A

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)

28
Q

How do we perform the ADF test?

A

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)

29
Q

How do we perform an ADF test with drift and deterministic trend?

A

∆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

30
Q

What is the main issue with an ADF test?

A

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)

31
Q

What alternative tests are there?

A

KPSS, Phillips-Perron, Modified ADF test

32
Q

What is unique about the KPSS test?

A

It is the only test listed in which the null hypothesis is stationarity

H₀: Stationarity

H₁: Unit root

33
Q

How does the KPSS test work?

A
  • 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
34
Q

Why is KPSS a stronger test?

A

It helps to mitigate the ADF power problem in small samples

35
Q

How does the Phillips-Perron test work?

A

∆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
36
Q

What is the decision rule of the Phillips-Perron test?

A

Decision rule: reject the null if test statistic lower than critical value

37
Q

What is the drawback of the Phillips-Perron test?

A

It tends to perform less well than ADF in finite/small samples

38
Q

How does the Modified ADF test (DF-GLS) work?

A

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)

39
Q

What is the decision rule of the Modified ADF test?

A

Reject the null if test statistic is lower than the critical value

40
Q

Do these tests work for multiple unit roots?

A

No, so we need to difference to I(0) to make it stationary

41
Q

How does forecasting with a random walk with drift [x(t)=α+x(t-1)+ε(t)] differ from forecasting with a stationary series?

A

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

42
Q

How does forecasting with a trend stationary process [x(t)=α+βt+ε(t)] differ from forecasting with a stationary series?

A

E[x(T+k)|I(T)] = [x(t)-ε(t)]+βk ←-ε(t)+βk

V[(x(T+k)|I(T)] = σ² ←no change

43
Q

What is seasonality?

A

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)

44
Q

How do we deal with seasonality?

A

1.Try to check for this pattern while looking at the differences

45
Q

How do we fix a model with pure seasonality?

A

We can estimate the model with only the 4th lag i.e. x(t)=θ₄xt-4+εt

46
Q

What is the issue with this method?

A

View pic

47
Q

How do we fix a model with a mixture of seasonal and non-seasonal?

A

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)

48
Q

What alternative approach can we use?

A

Introducing a multiplicative seasonal factor

  • this implies an interaction between seasonal and non-seasonal parts

Econometrics (10 decks)

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