Brooks remainign

Question 1

why is it important to test for non-stationarity

Answer 1

for one, if we have a non stationary time series, the effects of hte hsocks will not diminish. This means that the shocks seen previously still influence current levels. The outcome of this is that the time series will likely explode.

Question 2

elaborate on types of non stationarity

Answer 2

1) random walk with drift
2) trend-stationary process

Random walk with drift is given as:

r_t = mu + r_{t-1} + u_t

Trend-stationary process is given as:

y_t = a + b t + u_t

Question 3

elaborate on the trend-stationary process

Answer 3

It is an interesting time series because it is stationary around a linear trend.

It is also commonly referred to as “deterministic non-stationarity” and it needs to be detrended to work with it.

Question 4

non stationarity refers to everything that is not statioanry. but typically, with non-stationary one will consider the unit-root non stationary ones. Why?

Answer 4

Because there are very few use cases for the non-stationary ones, but the ones with a unit root is an important field.

With ones that are not unit root, but still non-stationary, the shocks will increase in effect as time pass by.
With unit root, the shocks remain as they were, we basically get a sum of them.

Question 5

how do we de-trend a deterministic trend-stationary time series?

Answer 5

we run the following regression:

r_t = a + b t + u_t

Then we use the residual series as the new variable. it should now have the trend removed.

Thi sassumes that the trend statioanrity was indeed the one as given

Question 6

what is stochastic non-stationarity?

Answer 6

Another term for hte random walk with drift.

There is a stochastic trend in the data, and we know this because the next value is equal to the previous value plus error term. if the error term has the assumption of mean 0, we get increasing variance as we move further out.

Question 7

how can we make a stochastic trend “stationary”?

Answer 7

First difference. For instance, if we have random walk, performing the first differnece on it will give us a statioanry sequence.

Question 8

what is removed when we use differencing on stochastic trend?

Answer 8

We remove the previous value. this leaves the trend and the error term a_t.

it is impoirtant to understand that the trend is included in the difference.

Question 9

what happens if we first difference a trend stationary time series?

Answer 9

We get an MA(1) case

Question 10

difference between de-trending and differneicng?

Answer 10

in this case, de-trending refers to performing regression and then using the residuals. It works on the determinsitic trend model, but not on the stochastic trend.

de trending removes the overall trend.

Differencng removes the stochastic trend, but not their individual effects. This is why differencing is great. We keep the shocks, but remove the trend. The outcoem is stationary.

Question 11

there is a crucial part about dickey fuller

Answer 11

it only work if the residuals are truly white noise.

Question 12

what to do if we want to test for unit root but our residuals have autocorrelation

Answer 12

Use Augmented Dickey fuller

Question 13

when is dickey fuller valid+

Answer 13

when u_t behaves as white noise

Question 14

under what circumstances can Dickey Fuller be weak?

Answer 14

if there are structural breaks

Question 15

what happens if we combine variables X_it that have different orders of integration?

Answer 15

They receive integration equal to the one with the higest order

Question 16

what does it mean to be cointegrated?

Answer 16

if a linear combination of variables that are all I(1) become I(0) together, then they are conintegrated.

Question 17

what does it mean to be cointegrated?

Answer 17

there will be a long run equilibrium relationship between tehm

Question 18

elaborate on the definition of cointegration by Engle and Granger

Answer 18

A set of variables are cointegrated if there exist a weight vector alpha that when taking the dot product of alpha and the variables (linear combinaiton) we get a series that is I(d-b), which when all are I(1) produce a time series with linear combination I(0)

Question 19

why not just first difference each time series, and work from there?

Answer 19

This is correct for ARMA process. However, if we want to leverage common effects between the variables, we need a differnet approach.

Question 20

is univariate time seires models sufficient here?

Answer 20

No, we need multivariate.

Question 21

give example of multivariate time series model

Answer 21

y_t = ax_t + u_t

y_t could be the forward price, and x_t could be the stock price. this allows us to model changes in forward price as a funciton of the stock price.

Question 22

what is special about y_t = a x_t + u_t

Answer 22

It consists of 2 variables that have a unit root each.

Question 23

consider the model:

∆y_t = b ∆x_t + u_t

elaborate on it

Answer 23

It is a first differneced on both sides.

If it reach a long run equilibrium, this shit wont captiure it becasue the differencing eventually just creates 0 = 0.

Therefore, first differencing is not good to capture long run equilibriums

Question 24

what do we call the model that can be used for these kinds of cases?

Answer 24

Error correction model

equilibrium correction model

Question 25

give the shape of a error correction model

Answer 25

The part that will be replaced by a placeholder variable, (y_{t-1} - gamma x_{t-1}) is called the "error correction term".

Question 26

how do we find the "cointegration coefficient"?

Answer 26

from the cointegration regression

Question 27

elaborate more deeply on the differnet aspects of the "error correction model" in the bivariate case

Answer 27

We have: ∆y_t = b_1∆x_t + b_2 (y_{t-1} - gamma x_{t-1}) + u_t There are 2 components: 1: Long run 2: Short run beta_1 describes short run relationship. beta_2 sescribes teh speed back to equilibrium gamma describes the long run relationship.

Question 28

What is the "granger representation theorem"?

Answer 28

The granger representation theorem say that "if there exists a dynamic linear model with stationry disturbances and the data is I(1) ,then teh variables must be cointegrated of order (1,1)." Interpretation: I(1) imples that they become stationary after being differenced once. if there exists a model that makes teh residuals stationary, THEN: it guarantees that we can model using Error Correction Model of discussed shape, and hte cointegration combination is I(1-1)=I(0)

Question 29

recall what it means that two variables (or more) are cointegrated?

Answer 29

They share a long run relationship

Question 30

what does it mean to share a long run equilibrium relationsip?

Answer 30

if two variables share a long urn equibrlium relationsihp, meaning that they are cointegrated, it means that although they will likely drfit etc, they will never drift far away from eahc other. They will follow the same tendencies and be correlated in some way.

Question 31

how can we check whether a bunch of variables are conintegrated or not

Answer 31

We obtain residuals, and then run Dickey Fuller on the residuals. However, due to the difference in regards ot now being used on residuals, there is a new set of critical values, and it is commonly referred to as Engle-Granger test.

Question 32

assume we have a bunch of variables that are non-stationary, and we susepct that they are cointegrated. What to do?

Answer 32

we present 2 primary methods: 1) engle-Granger 2-step 2) Johansen

Question 33

elaborate on Engle-Granger 2-step

Answer 33

2 steps, obviously: 1) Make sure that each variable is I(1). Run the conintegration regression using OLS. Check the residuals for stationarity. If they are I(0), proceed to step 2. 2) Set up the new regression, which has the shape: ∆y_t = b1 ∆x_t + b2 (residual_{t-1}) + v_t the residuals are just the residuals from the "arbitrary" regression performed in step 1. Can include constant, can choose not to etc.

Question 34

how do we decide on which variable to choose as y_t, and x_y (LHS vs RHS)? How about when we have many variables that are all I(1) and we susepct is I(0) when combined?

Answer 34

The important thing is that our choice does not affect the fact that the bunch of variables are cointegrated. therefore, our choice depends on what specific relationships we are interested in exploring. If we want to see how changes in x_t affects the long run eq of y_t, then we model it as: y_t = x_t .... we can easily do the reverse, but then we are looking at how y_t affects x_t.

Question 35

elaborate on the validity to perform inference during Engle-granger 2-step

Answer 35

the first regrssion holds no meaning. but the second is free for interpretation