QM Week 10

Whats dodds hazard rate. What does it depend on?

(dp/dt)/1-pt It depends on bp

Whats mans fields hazard rate. What does it depend on?

(dy/dt)/K-Y It depends on b(Y/K)

If we regress rw1 on rw2 what do we expect B0 to equal?

0

If we regress yt on yt-1 what do we expect B1 to equal?

1

The dickey fuller test is a ... tail test

1

Compare white noise to a random walk (2)

1. Both have movements that are purely random and unpredictable 2. White noise level at any point DOES NOT rely on past history WHEREAS random walk level at any time DOES depend on past history (i.e. White noise is not trended but RW are)

WHy might we be interested in whether economic data behave like a random walk (3)?

1. Speculation in financial markets- would we really want to to be on a share if its price follows a random walk 2. Forecasting- if the variable is a RW, we need to incorporate that information into the model we develop 3. Modelling and testing hypothesis. The tests are different when we are modelling a random walk- t-tests don't work

What is the consequence of the fact that the traditional t-tests are only useful for stationary series and not non stationary series like random walks (2)

1. how do we test if a series is a RW? 2. If our data is a time series and we do a regression of y on x, how do we know the results aren't spurious?

Problem 2- how do we know if our results are spurious when we regress x on y?

1. problem only occurs if time series data is non stationary 2. use dickey fuller test to see if unit root/RW. 3. if the results show that it is a RW, the t-test results may be spurious

Define a simple random walk

A random walk is a purely random and unpredictable time series whose level at any time depends on past history yt-y(t-1)=errort

What is a stationary series. Give an example of one

A time series whose statistical properties do not change over time. White noise is one of many stationary series

What is a non-stioanry series. Give an example of one

Basic statistical properties are not constant over time. Instead they change over time. E.g. random walk

Why is it that, if we do a regression of the form yt on yt-1, where y is a random walk, we would expect to find that the slope coefficient is not significantly different from one? B1=1

Because if it is a simple random walk, we have yt=yt-1+ℇt or if u like yt=B1yt-1+ℇt where B1 must equal 1

Why isn't all stationary series white noise?

Because stationary series can have autocorrelation (as long as it is constant) but white noise is never autocorrelated.

If we regress a random walk on another, why must the coefficient B1=0

Because, by contraction, two random walks must be independent of each other

How to get around problem 1- how do we know if our time series is a random walk?

Do a dickey fuller test

What is the null and hypothesis for dickey fuller

H0: has unit root HA:does not have unit root

Why is a dickey fuller test better at not letting us reject the null that B1=1

It has higher critical values for each degree of freedom

What is spurious regression

Regression results look good but in fact have no meaning (t-tests on non stationary series)

What is the decision rule for Dickey fuller test

Reject null if absolute value of the t-stat |t-stat|>|critical value| (absolute value of CV)

Define the first difference

Show equation then say "the change between the variable between two points in time, The change is white noise with a mean of zero"

"For a simple random walk E(yt+1) = yt". What can we say about E(yt) where y is a simple random walk? Imagine we are in period t-1 and we have to forecast yt, what would our forecast be?

The forecast for yt+1 will be a forecast of the value of yt + εt. Now εt is purely random with a mean of zero. So the expected value of yt+1 will be the expected value of yt + εt and E(yt + εt) = yt + 0 = yt.

If we know that we have spurious results, how can we get around it and still test whether x is related to y? Whats the logic behind this

We can do the regression on the first differences rather than on the levels because even if the variable itself is a (non stationary) random walk, the first difference will be stationary (because the change between two variables is WHITE NOISE!)

Define a random walk with drift

Where the time series for the change in y between two periods has a mean which is not zero

define White Noise

a purely random series which has a constant mean (usually zero) and constant variance. There is no correlation or trend

If variable is nonstionary, is the first difference of the variable stationary/non stationary

it will be stationary

Can white noise have autocorrelation

no

Normal t-tests of H0:B1=0 HA:B1≠0 and H0:B1=1 and HA:B1<1 are not useful for what type of time series

non stationary

another word for trended data/data with a long term trend is...

non stationary series

A unit root is also known as a

random walk

Equation for random walk with drift. Rearrange to show level of y at any given point

yt-y(t-1)=d+ℇt yt=y(t-1)+d+ℇt

Equation for simple random walk. Rearrange to show level of y at any given point

yt-y(t-1)=ℇt yt=y(t-1)+ℇt