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METRICS TERM 2 > HANDOUT 3 > Flashcards

HANDOUT 3 Flashcards

1
Q

3 conditions for weak stationarity

A
  1. E(Yt) = M
  2. V(Yt) = sigma^2 Y
  3. COV(Yt, Yt-h) = gamma h
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2
Q

Does AR1 satisfy condition 1 for weak stationarity? Give E(Yt) when we include a mean.

A

YES

E(Yt) = M / (1 - phi)

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3
Q

Does AR1 satisfy condition 2 for weak stationarity? Give V(Yt) when we include a mean.

A

V(Yt) = sigma^2 / (1 - phi^2)

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4
Q

Does AR1 satisfy condition 3 for weak stationarity?

A

YES
Pj = phi ^j
As long as phi<1, also satisfies weak dependence.

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5
Q

Therefore, what condition do we need for an AR1 process to be stationary?

A

Phi<1 in absolute magnitude

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6
Q

A process which is stationary is integrate of order…

A

I(0)

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7
Q

I(1) means

A

Integrated of order 1

The first difference of the series is stationary

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8
Q

How many unit roots does as I(1) series have?

A

1

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9
Q

How many unit roots does an I(2) series have?

A

2

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10
Q

Random walk model formula (with drift)

A 
Yt = M + Yt-1 + €t
Phi = 1
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11
Q

Yt = when we back substitute a random walk model

A

Yt = Mt + Y0 + sum €j

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12
Q

E(Yt) for a random walk model. Does this satisfy stationarity?

A

E(Yt) = Mt + Y0

E(Yt) depends on t = not stationary

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13
Q

sum €j refers to

A

Stochastic trend

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14
Q

Mt =

A

deterministic trend

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15
Q

V(Yt) for a random walk model. Does this satisfy stationarity?

A

V(Yt) = t sigma^2

NO - uncertainty increases over time

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16
Q

Ph = for a random walk model

A

Ph = (t - h) / t

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17
Q

In theory, ACF and PACF for random walk

A 
ACF = horizontal line at p=1
PACF = spike for 1 at p=1, others 0
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18
Q

In practice, ACF for random walk

A

Finite t

Ph decays at a LINEAR rate

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19
Q

random walk without drift

A

M=0

Stochastic trend only

20
Q

random walk with drift

A

M≠0
Stochastic trend AND deterministic trend
ACF shows even less decay to zero.

21
Q

The test for unit roots / non-stationarity is called

A

Augmented Dickey-Fuller (ADF) test

22
Q

Easier method for ADF test (for p*=0)

A

change Yt = gamma Yt-1 + €t

gamma = phi - 1

23
Q

H0 and H1 for ADF test

A

H0: gamma = 0 - non-stationary
H1: gamma < 0 - stationary

24
Q

ADF test is one/two sided?

A

ONE sided - the root is either unity or less than.

25
Q

Test stat for ADF test

A

Do a t-test

26
Q

Problem with CVs for ADF test

A

We test under H0 for non-stationarity so the distribution of the test stat is shifted left compared to t/z test = more likely to reject H0.

27
Q

Solution to CVs for ADF test

A

Use Mackinnon’s CVs table 10

CVT = phi infinity + (phi1/T) + (phi2/T^2)

28
Q

When do we reject H0 for ADF test?

A

1 sided test so only reject if test stat < CV.

29
Q

What is the biggest root of an AR?

A

Biggest root = sum of all parameters

30
Q

gamma for AR2 =

A

gamma = phi 1 + phi 2 - 1

31
Q

For an AR(P), test equation =

A

change Yt = gamma Yt-1 + sum j=1,…,p*

delta j change Yt-j + €t

32
Q

state 3 ways of choosing P* for ADF test

A
  1. Look at number spikes on PACF
  2. Minimise a selection criteria
  3. Test between different models
33
Q

Max P* =

A

T^1/3

34
Q

How do we test between P=3 and P=2?

A

Test H0: delta 3 = 0 i.e. is the coefficient on

change Yt-3 significant? If we do not reject H0, then test between P=2 and P=1.

35
Q

As well as choosing P*, what else do we have to decide on for the ADF test?

A

Deterministic elements e.g. do we include an intercept and/or trend?

36
Q

Model A =

A

no intercept, no trend

37
Q

Model B =

A

intercept, but no trend

38
Q

Model C=

A

both an intercept and a trend

39
Q

When should we use Model A? explain.

A

NEVER
Under H0, E(Yt) = Y0
Under H1, E(Yt) = 0
Only reconcile by assuming Y0=0 under H0 but unrealistic to assume the initial value of any variable is zero.

40
Q

When should we use Model B? explain.

A

Under H0, E(Yt) = Mt + Y0 - time trend
Under H1, E(Yt) = M / (1 - phi) - constant
To reconcile assume M = 0 under H0
Only good if realistic to assume zero average growth rate e.g. ER, IR, inflation

41
Q

When should we use Model C? explain.

A

Most of the time
Under H0, E(Yt) = Mt + Y0 + alpha t(t+1)/2
Under H1, E(Yt) = M/1-phi + alpha/1-phi t
Assume alpha=0 under H0
Realistic - the growth rate of a variable doesn’t usually following a trend.

42
Q

Power of ADF test

A

LOW POWER = we often do not reject H0 = often find non-stationarity when actually series is stationary.

43
Q

Perron’s result

A

A stationary series with a structural break (in intercept/trend) will be found to be non-stationary in an ADF test.

44
Q

If we do not reject H0, how can we test to see if the series is I(1) or actually I(2)?

A

change^2 Yt = M + gamma changeYt-1 + sum
delta j change^2 Yt-j + €t
H0: gamma=0 - series is I(2)
H1: gamma < 0 - series is I(1) as 1st difference is stationary.

45
Q

Are many economic series I(2)?

A

Not many - maybe Venezuelan prices

NO economic series > I(2)

METRICS TERM 2 (6 decks)

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