NonStationarity Flashcards by Phil Moore

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

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What is the formula for a random walk process?

x_t = x_t-1+ε_t (note: this is an MA process where θ=1)

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

Yes

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Is the Variance constant?

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

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What is an integrated process of order 1?

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

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How do we difference a process?

We form the equation with ‘L’ and factorise this

We note that (1-L)y_t = y_t-y_t-1 = Δy_t

i.e. x_t=1.3x_t-1-0.3x_t-2+ε_t

(1-1.3L+0.3L²)x_t=ε_t

(1-L)(1-0.3L)x_t=ε_t

Δx_t=0.3Δx_t-1+ε_t

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How do we represent the second difference?

(1-L)²y_t=Δy_t-Δy_t-1=Δ²y_t

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

Mean and Variance

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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

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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)

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What is the value of z(t)?

z(t) = x(t) - βt

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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

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How do we make a random walk with drift stationary?

Through differencing Δx(t) = α + ε(t)

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

x(t) = α + βt +x(t-1) + ε(t)

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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

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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)

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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

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How do we determine a random walk process through correlograms?

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

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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

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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 x_t-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 \< c_DFtrend 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 x^d_t=x_t-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)] = **k**σ² ←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)=θ₄x_t-4+ε_t

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. x_t = θ₁x_t-1+ θ₄x_t-4 + ε_t -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