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EC 511 Quants > Term Two > Flashcards

Term Two Flashcards

1
Q

What is a time - series?

A

It is an ordered sequence of data where the order refers to increase dates denoted by time t.

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2
Q

What are basic concepts of time series which are important?

A

Stationarity, autocorrelation, multivariate

They are used to forecast and analyse policy.

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3
Q

What are the sylized facts of economic time series?

A

1) most series contain some form of trend.
2) Some series seem to meander, they have what is known as a random walk. They have a stochastic element.
3) Shocks to a series will display persistence, it takes a long amount of time to subside effects.
4) The volatility of some series will change over a time period.
5) Some series will share a co-movement with another series.

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4
Q

What is the concept of persistence of a time series?

A

What it means if; if there is a small shock to your data, it will take a very long time for the effects of the shock to die out.

In time-series, the stochastic process is an ordered sequence of random shocks to the data.

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5
Q

How can you detect for persistence?

A

By using an autocorrelation plot.

An autocorrelation plot is a histogram which shows the value of autocorrelation Rho h.

Between one data point and a data point lagged h periods previous.

The data reading shows the autocorrelation between one data point and another lagged one previous period.

e.g; if it is 0.035 it the correlation between Y and Yt-1 is 3.5%.

The longer high autocorrelation lasts for, the more persistent the data set is.

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6
Q

How do you work out the standard error of autocorrelation rho h values?

And therefore test if rho h = 0

A

SE = square root of (1/T)
Where T is the value of the number of observations in the data.

We say that the time series is purely random then the auto correlation coefficients are rho h = 0.

From this you can do a standard confidence interval.

If rho h = 0 is within the interval, we do not reject the hypothesis that the autocorrelation could be rho h = 0

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7
Q

what are the formulas for Q stat and LB stat ?

A

Q = T sum of Rho h^2 from m to h=1

Where m is the length of the lag and T is the number of observations in the data.

LB stat = T(T+2) sum of (Ph^2 / T-h) between m and h =1

they test the joint hypothesis that all rho h up to a certain lag are simultaneously equal to zero.

Use the CHISQ tables to get values

Degrees of freedom is the number of rhos that you sqaure.

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8
Q

What is a stochastic process?

A

It is an ordered sequence of random variables.

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9
Q

What is a partial realization of a stochastic process.

A

It is a small section of a larger process.

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10
Q

Define a stationary stochastic process.

A

It is said to be stationary if the mean and the varience of the process are constant over time.

(do not depend on t)

The covarience between the two time periods is dependent on only the distance of the time gap, not the actual value of the time.

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11
Q

Why do we need the autocorrelation to be independent of time?

A

because this will show that the part of the series we are analysing is not too dependent on the past.

This allows is to infer things about the whole series that is observed.

We say that autocovarience and auto correlation are not a function of time.

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12
Q

What will the autocorrelation correlelogram show for a stationary time series?

A

The histograms and values will be low. They will be much higher for a non-stationary time series.

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13
Q

Can a stationary time series have any persistence?

A

It can have some short term persistence which will die out within the time of a few lags.

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14
Q

Define a time series process.

A

It is a finite realization of a random process in discrete time periods.

Collection of random variables which are indexed by t.

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15
Q

What is autocovarience?

A

it is a measure of the link between observations at different points in time.

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16
Q

What is a white noise process?

A

it is the most simple stationary stochastic model within econometrics.

it has mean 0 and constant varience.

Homoscedastic random variables.

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17
Q

Stationarity holds when there is a constant difference between the Xt and Yt variables.

A

the mean of the two is constant level apart for whole series.

Does not depend on t.

The variance of both xt and yt is also constant through time.

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18
Q

What is an autoregressive process?

A

Is a process that acts under the premise that past values will have an effect on current values.

AR(1) is a first order. The current value is based on the value from the previous period.

AR(2) is a second order. The current value is based on the values of the previous two periods.

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19
Q

When you have Co as 0 what can you use recursive substitution to do?

A

You can sub in previous periods until you get;

Yt = B1^hEt-h +B1Et-1 +et

It will be stationary if when

B1^h approaches 0, h approaches infinity.

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20
Q

What do we know about the value of B in AR(1) processes?

A

We know that if;

|B|

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21
Q

Explain the thinking behind the lag operator.

A

It will transform one time series to another by lagging it by one period in time.

Lyt= yt-1

L^nYt = Yt-n

it only works on variables which contain a constant.

L^0 = 1

Always.

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22
Q

Derive the mean of an AR(1) process.

A

You do it with Yt=B1Yt-1 + Et

B1 can be used to get to yt with a lag and factor out the lag

MEw= E(et)/E(1-b1)

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23
Q

What is gamma 0 denotes its?

A

Variance. (most particularly the variance of an AR(1) process).

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24
Q

What is meant by the expected value of a sequence?

A

The expected value of a sequence effectively means the long-run average value of that particular variable.

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25
Q

Write out the derivation of;

  • mean
  • variance
  • covarience
  • autocorrelation

of the process.

Yt = Co + B1Yt-1 +et.

Yt-h for covarience and autocorrelation.

A

Seminar 2 answers in folder.

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26
Q

What is the stationarity condition for an AR(2) process?

A

|B1+B2|

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27
Q

What does ADL model mean or ardl?

A

Autoregressive model with a distributed lag variable.

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28
Q

What statistics do we calculate in order to select the best fitting model for the data?

A

Akaike (AIC)

and Schwarz for each model.

The model with the lowest AIC and Schwarz is selected as the one which is the best fitting.

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29
Q

How do you calculate the AIC and Schwarz information?

A

C= The sum of squared -2( residuals (RSS)_ + penalty term c(k,t)

function of K,T.

The penalty term differs for AIC and Schwarz.

AIC ( 2k)

Schwarz k log T

k = no.of parametres and T is the number of observations in the data

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30
Q

How can you forecast an AR(1) process one step ahead?

Then two

Then three

A

Yt+1 (hat) = F(Yt+1 | It) +F(et+1 | It)

Where It is all of the information that you have available to you at the time t.

Forecasting the error term in the future will always be zero because there is no way to predict what will occur in the future.

Two and three step ahead are effectively the same process.

Yt+2 = B1^2Yt

Yt+3 = B1^3Yt

So H steps ahead;

Yt+h =B1^hYt

B1^h - 0 h - infinity

If there is a Co not dependent on time, subtract it from the question and then add it back later.

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31
Q

What is the forecasting error?

A

It is a confidence interval which you will find the forecast to be between.

Yt+1 -Yt+1 hat = Et+1

Yt+h - Yt+h hat = et+h +B1et+h-1 +…+B1^h-1et+1

You then calculate the standard error, sigma squared and then square root that.

sigma sqaured e time sum of B1^(2j) between h-1 and J=0.

From these figures, you can construct a confidence interval.

Yt+h - c1-gamma/2 (sigma)

and then Y1+h + the above

C1-gamma/2 is the 100 percentage point of the standard normal distribution

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32
Q

What does VAR series stand for?

A

Vector Auto regressive Process.

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33
Q

In a VAR series is there any cross-correlation?

A

No e1,t x e2,t =0

34
Q

What is the expected value of the two errors?

A

0 and 0 expressed in a matrix format.

35
Q

What is the expected value of E(etet-k)?

A

It is R k = 0 and 0 if k is not equal to one.

Where R is sigma 2 e1 , 0

0, sigma squared e2

Written in matrix format

right to left.

36
Q

Write out a simple VAR model.

A

Lecture week 16 slide 8.

Matirx form;

with y1,t and y2,t

Then b11,B12,B21,B22

Then times by

y1,t-1 y2,t-1

Then plus e1,t e2,t

Given by Yt=AYt-1 +Et

37
Q

What happens if the eigen values of A are less than one in absolute value?

A

As lim of n approaches infinity

A ^n = 0

When you have used VAR recursive substitution to go back to

Yt=Et +AEt-1 +…+A^nEt-n+…

38
Q

how to calculate the eigen value formula?

What is an alternative method?

how do you know if the process is stationary?

A

You use lambda identity matrix and minus them.

Then take the mod of the matrix.

ad - bc

Then once you have equation;

Use the quadratic formula to solve.

ALTERNATIVE:

A^k

you can multiply the matricies together, remembering the rules!!

The process is stationary when the eigen values are in between -1 and 1

_1

39
Q

In what way can the VAR be estimated?

A

You can estimate the var by using OLS regression.

You multiply out the vectors for each series and then call them

ARDL(1,1) for y1,t

ARDL (1,1) for y2,t

40
Q

What are the advantages of the VAR model?

A

Simple OLS estimation

Methodology is simple.

The forecasting is fairly reliable

41
Q

What are the disadvantages of the VAR model?

A
  • It is a-theoretic the information used prior is less.
  • VAR’s are not good at policy analysis only forecasting.
  • All variables must be stationary.
  • Difficult to estimate if you have a large number of variables.
42
Q

What is the VAR forecast for;

  • One step ahead
  • H step ahead
A
  • One step

Yt+1 hat = A1hat YT

Yt+h hat = A^h1 hat Yt

43
Q

What does it mean if something Granger causes something else?

A

It means that one variable within a regression will have another effect within the regression.

Hypothesis is

Ho: B12 = x12 = 0
H1: B12 not = x12 not = 0.

Then you just run an F-test on this

Given the model;
y1,t =b11yt,t-1+b12y2,t-1 +x11yt,t-2 + x12y2,t-w +et

44
Q

What does it mean if a trend is deterministic?

A

It means that the trend is always the same, no matter what.

45
Q

If we have a process, such as;

yt = Co+Bot+et

Which parts are stationary and where is the deterministic trend?

A

yt is non=stationary

Co+Bot is deterministic trend

et is the stationary White Noise Process.

46
Q

In a trend stationary model, what effect will a shock Et have on the trend?q

A

It will have no effect on the trend part of the model.

it will persist for quite a while do to its effect on the autoregressive aspect of the model.

47
Q

If we have the model

Yt = Co+B0t+Ut

where Ut = B1Yt-1 +et

Why is Ut called Ut?

Break it into deterministic trend and autoregressive portion.

A

Co + Bot is the deterministic trend.

B1Yt-1 +et is the autoregressive part.

it is called Ut because it is a persistent process.

48
Q

what is the case of OLS inference with deterministic trend?

A

OLS inference tools remain valid when linear trend is included.

Deterministic trends do not capture the non-stationary part of the economic series.

We need more flexible trends

We cannot rely on the usual t-test.

49
Q

How can you detrend a trend stationary model ?

A

Firstly write the model out in a matrix format.

then find the residuals ;

ut hat = Yt -Co hat - Bot hat)

Then Given these residuals, specify an AR model which you select using the AIC and Schwarz criteria.

And fit this model to the residuals Ut hat.

50
Q

How to forecast a trend stationary process?

A

DETERMINISTIC PART
-For the deterministic aspect;

The forecast is the same, as it will always be on trend no matter what.

AUTOREGRESSIVE PART:

Ut+h hat = B1^hUt

Because you cannot forecast the epsilon part of the model.

PUTTING IT TOGETHER:
Yt+h hat = Co + Bo(T+h) + B1^h(Yt-Co -Bot)

Because that is the value of Ut

Don’t forget the forecast interval and confidence error.

51
Q

What is a random walk process?

A

If a process has a high level of persistence it will take a long time to revert to it’s mean after a shock.

It is said to ‘Walk at Random’.

Yt = Yt-1 +et is an example of a random walk process.

It is an AR(1) process where B1 = 0 and C0 = 0.

There first difference of this process is stationary.

This means it is Integrated of Order one process. I(1)

Yt - Yt-1 = et which is stationary

52
Q

What is the efficient capital market hypothesis?

A

It is that stock prices are random and there is no way to predict them.

53
Q

What is the intuition behind a random walk model?

A

It is completely random, the value at time t is the value at time t-1 plus a completely random error.

54
Q

What happens when you use substitution on a random walk model?

A

Yt = Et + Et-1 +…+E1

Where E1 is the initial shock to the market.

Could also be;

Yt = Et + B1Et-1 +…+ B1^hEt-h

It can be written as Yt = the sum of shocks (ei) between t and i = 1

55
Q

What is the effect of a shock Et-h on a random walk series?

A

It has an infinite effect. It does not die out at all.

There are no autoregressive coefficients to smooth out its effects.

56
Q

What is the autocovarience of a random walk process?

A

Cov(yt,Yt-1)

= E[(et+et-1+…+e1)(et-1 + et-2+…+e1)]

(t-1)sigma squared.

Cov(Yt,Yt-j)

(t-j)sigma squared.

57
Q

What is the autocorrelation of a random walk?

A

Cov(yt,yt-J) / SQUROOT (varYt)(varYt-j)

=t-J/SQUROOT (t(t-j)) =
1

58
Q

What is true about RW and OLS?

A

OLS does not hold for RW functions.

t-stat for B1 = 1 is not distributed normally.

59
Q

What is the substitution of RW with drift?

A

RW with Drift;

Yt = Co + Yt-1 + et

Yt = Co + (Co +Yt-2 +et-1)+et.

…

Yt = Cot + Et + Et-1 +…+ E1.

60
Q

What is the mean of a random walk with drift process?

A

Expected value of all the errors which is all zero.

Also the expected value of Cot

Which means the mean is Cot.

61
Q

What is the effect of a shock in a random walk?

A

It will be extremely persistent. It will never factor out.

62
Q

What is the forecast of a random walk process?

A

You know all of the values up to T.

Yt+1 bar = Yt + F(Et+1| It)

But forecast of the error is zero.

Yt+1 bar = Yt

Forecast is just the previous value as it is impossible to forecast a random shock in the future.

Even H step ahead;

Yt+h bar = Ybar

because it is impossible to forecast the shocks.

63
Q

How to calculate the forecast error of a random walk variable?

A

yt+h - Yt+h bar = Et+h + Et+h-1 +…+ Et+1.

sigma sqaured = E(Yt+h - Yt+h bar)^2)

Sigma ^2 times sum of 1 h-1 and i = 0

Therefore Sigma^2 (h-1)

is the forecast interval.

The interval grows without bounds.

This prevents us from having a credible confidence interval for the forecast.

64
Q

What is the mean and Var of A RW with Drift?

A

Expected Value Yt = Co

Var = sum (Et-i)

Tsigma^2.

Because the other stuff has no variance.

65
Q

Why would we test for a unit root?

A

Because it allows us to see if the process is TS or DS. This will allow us to know the effect that a shock will have on the series.

66
Q

What is the unit root test?

A

If we have the equation;

Yt=B1Yt-1 +et

Then we subtract yt-1

We get

Change in Yt =Rho yt-1 +et

Where Rho (P) = B1-1

Ho P=0 and stationary

H1 P

67
Q

What are the three observations of the unit root test?

A

One:

The distribution of the t-test is not completely normal.

Two:

The process can be a random walk with a weak dependence process as well.

Like an AR(p) memory process.

We fit a short memory process and run the ADF test.

Three;

The RW may be moving around a deterministic trend.

We must test for these.

ADF test becomes;

change in Yt = Co + Bot +(B1-1)Yt-1 + Yt-1 +et.

If it is stationary it will be;

Yt = Co +bot + B1Yt-1 +et

If not then you drop the B1.

68
Q

What are the critical values of the DF test?

A

Alpha 1%

CV = -3.43

Alpha 5%

CV = -2.86

Alpha 10%

CV = -2.57

If the P value is smaller than the Alpha value then you should reject Unit root and accept H1.

69
Q

What is the ADF test?

A

Where there is an AR part to the equation. and an error. The persistence is model in the AR part of the process.

Derivation on lecture wk 20 pt 1 slide 30 LEARN IT

70
Q

How do you decide how many lags for the ADF test?

A

You select the model with the lowest AIC and Schwarz value available.

Ensure autocorrelation is ruled out. If not choose one with more lags.

71
Q

When would you run ADF?

A

Larger data sets

When there is more than one lag.

72
Q

What do you need to remeber to include when deriving the ADF test?

A

Make sure to include the rule of thumb (on your Whiteboard)

73
Q

What is the Heuristic Rule for Spurious Regression?

A

It is where the R^2 >DW stat.

This means you have well fitting residuals but there is high correlation of the errors.

We say that the model spuriously fits the data.

74
Q

What is spurious regression?

A

When you run a regression on two unrelated processes and there seems to be a linkl.

  • High r^2
  • t-STAT says reject H0.

Consider that the larger the sample, the more likely you are to accept H0 as significant.

75
Q

What is the general rule for cointegration?

A

Xt - I(d)

Yt - I(b)

Zt = Xt +lambdaYt I(c)

C= max(d,b) d not equal b

C

76
Q

Give an example of a test for cointegration?

A

If you have yt and Xt and you find B a parameter.

If Bxt has constant distance from Yt, your process is said to be cointegrationed.

Then the residuals of the new regression should be stationary I(0).

B1Ut-1 then run a DF test H0:B=0 H1:B not = 0

H1 is cointegration H0 is spurious regression.

Compare with DF2 CV’s as they are more stringent.

77
Q

Is Spurious regression the end of economic analysis?

A

No;

1) Cointegration to see if there is a cointegrating factor.
2) ADF test to see the order of the cointegration
3) The data could be remodelled using ARDL.

78
Q

What is the ECM representation?

A

Yt = AYt-1 +Et

This is a VAR representation.

SUbtract Yt-1 to get;

Change in Yt = |-|Yt-1 +Et

where |-| = A-1

This is the multivariate counterpart of DF test.

79
Q

What is the rank of a matrix?

Give all details of the columns independence

A

The rank of the matrix is given by the number of independent columns present.

A column is independent if;

  • One non zero element.
  • Cannot be identical to another
  • Should not be proportional to another
  • Should not be a linear combo of another column.
80
Q

What are the steps for the Johnson test?

A
  • Specify the model or add lags to the model using:

Sum of gamma i change in Yt-i between p and i=1

Step 2:

Calculate the rank of the cointegration.

If pie has n columns the model is invertable case 2

Test for n-1 then n-2 until it becomes significant.

n-k is cointegration rank.

n=0 is the random walk multivariate.

Step 3:

Output of the rank and maximisation will result in an estimation of B.

From this we can use OLS to find alpha.

81
Q

What are the Johanson test observations?

A

Cointegrating and ECM are a way to represent the long run equilibrium of two or more integrated variables.

You can only use the VAR forecasting model once it has been stationarized.

82
Q

What is the formula for auto covarience?

A

gamma h = B1^h gamma 0

EC 511 Quants (2 decks)

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