L5 - VARs and VECMS

Question 1

What is a VAR?

Answer 1

• Structural form
  
  • Contemporaneous interaction –> moving all the q_t and p_t onto one side
• Reduced form
  
  • Matrix created by moving the contemporaneous interaction matrix to the RHS and taking by taking the inverse

Question 2

What will the reduced form VAR errors look like?

Answer 2

Question 3

How does the system behave when subject to random shocks?

Answer 3

• backwards substitution like an AR(p) process
  
  • Subbed in for q_t-1 and p_t-1 , in the reduced form matrix (as the initial shocks arent multiplied by anything)

Question 4

example of an impulse response to a VAR system?

Answer 4

• matrix means:
  
  • middle matrix –> inverse of the contemptuous integration
  • last vector –> error term
    
    • Combined with the middle matrix it creates v_it
• Impulse response:
  
  1. Unit supply shock
     
     1. the significant initial impact on q, with a small negative impact on p
  2. Unit demand shock
     
     1. small impact on q, with a significant initial positive impact on p
• Impacts decrease over time till it converges to zero
  
  • Accommodated into the system

Question 5

Example impulse response graphs to a supply and demand shock?

Answer 5

• system only returns to its equilibrium position if they are stationary

Question 6

What is variance decomposition?

Answer 6

Ω matrix –>Var-Cov matrix of the structural disturbamces that are not correlated( don’t covary) but have their own variances

𝐶ε_t = v_t

• From below we have the matrics B₁ and C:
• To compute the variance decomposition
  
  • we assume that the variances of the structural disturbance are the same
  • so that the matrix Ω becomes an identity matrix

Question 7

Example of our variance decomposition after the Supply and Demand Shocks?

Answer 7

• Quantity
  
  • Use top number for Supply % –> SR impact
  • Use bottom number for Demand % –> LR impact
• Prices
  
  • Bottom number for Supply % –> LR impact
  • Top number for Demand % –> SR impact

Question 8

How can we go about estimating a VAR model?

Answer 8

lower triangular matrix means that we can definition compute the inverse

Question 9

Example estimation of a VAR model?

Answer 9

trend evolves closely together –> stochastic trend

• irf –> standard unit impulse function (code)
• bottom right- –> var-cov matrix for the residuals showing correlation
• GRAPHS: left impulse, right response
  
  1. shock of 1 unit to the Fed rate will take 8 periods to return to equilibrium
  2. impact on the 3 month t-bill rate is small when there is a shock to the fed rate
  3. With a shock on the 3 month t-bill rate, we see an increase in the fed rate over time –> it will cover/respond to other increases in the treasury bill (in the lR move towards 1)
  4. shock to the 3 month treasury bill rate is 1 and will remain there even in the LR

Question 10

How can we recover the structural parameters from a VAR model estimation?

Answer 10

Question 11

How do you estimate the Variance Decomposition?

Answer 11

Question 12

What is the Cholesky decomposition?

Answer 12

Question 13

Cholesky decomposition: sample moments and causal ordering?

Answer 13

Question 14

Example of using the Cholesky decomposition?

Answer 14

• upper triangular matrix –> assumes that fed rate is affected by both itself and the 3m t-bill rate
  
  • whereas the t-bill rate is only affected by itself

Question 15

How can we decide on the casual ordering of the variables?

Answer 15

• In the granger test, the left variables are dependent and the right are the independent
  *

Question 16

What is the general Granger Causality test then?

Answer 16

Question 17

What is the general form of the VAR (in matrix notation)?

Answer 17

• Only can find the reduced for if A₀ is invertible

Question 18

What is th solution to a general system of VAR?

Answer 18

• eigenvalues are needed to find out whether the system is stationary/stable
• cannot write a VAR system as a infinite moving average process if the system is not stationary

Question 19

Example of using eigenvalues and the unit circle?

Answer 19

Question 20

What do we do if a variable in a VAR is not stationary?

Answer 20

• In the discussion of VAR models above we assumed that the variables in the model should be stationary
• • What if one or more variables are non-stationary?
  
  • – standard statistical distributions not appropriate (results not valid)
  • – problems in interpreting the model and associated outputs
• • How to overcome this problem?
  
  • – make them stationary –> variables should be differenced
• • What if there are long-run or cointegrating relationships?
  
  • – rely on Vector Error Correction Models (VECM)

Cointegrating vector are the betas

Question 21

How are the outputs of VECM and VAR similar and different to each other?

Answer 21

• The output of a VECM is very similar to a VAR
  
  • – Impulse responses quantify response of variables to random disturbances
  • – Variance decomposition measures proportion of k-step ahead forecast variance that is explained by each of the random disturbances
    
    • • Both calculated in the same way as for the VAR
• • One important difference
  
  • – VAR with stationary variables –> impulse responses converge to zero
  • – Not the case for VECM
    
    • • if we write it in levels, where 𝑥 and 𝑦 are 𝐼(1), the transition matrix will contain at least one unit root
    • • random shocks to the system will have permanent effects on the levels of the variables

Question 22

Example of a VECM estimation?

Answer 22

• First coefficient estimates are the rate of adjustment parameter

Final conclusion –> if the BoE rate goes up by 1% the 10 year bond rate will vary by the same amount in the same direction –> WHY IS THE SIGN NEGATIVE THOUGH

Question 23

Example of an impulse response of the VECM estimation?

Answer 23

1. t-bill causes an intial increase to itself from the shock but returns t o1 and stays there forever
2. BpE rate will also tend to increase permanently over time due to a 10 year bond rate shock
3. After a BoE shock there is no initial change in the 10 year yields. But over time (after 2 periods onwards) they start to increase
4. BoE shock causes a slight increase in the BoE rate but afterwards, the shock persists into the future