VAR Model (k-variate VAR(p model (in the Reduced Form: where: Y t = A + B 1 Y t-1 + B 2 Y t-2 + + B p Y t-p + ε t Y t = (y 1t, y 2t,, y kt : a (k x 1 vector of time series variables A: a (k x 1 vector of intercepts B i (i=1, 2,, p: (k x k coefficient matrices ε t : a (k x 1 vector of unobservable i.i.d. zero mean error term (vector white noise (Note: the components of ε t are correlated, for the same t, in this reduced form representation of the VAR model. 1
On the simplest VAR model: Bivariate VAR(1 without intercept In the Reduced Form, this model is: [ ] where WN( Σ Σ [ ] 1. Derive the marginal sequences Method 1. Use the backshift operator B: [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2
Method 2. Do not use the backshift operator: [ ] Here, we assume and. Based on equations 3 and 4, we can also derive the expressions for and as shown below: After substituting } in 1 and } in 2 using equations 3, 4, 5 and 6, we can derive the following expressions: Finally, according to 7 and 8, the marginal models for } and } should be: 3
2. Derive whether the new terms for the marginal sequences } and } are white noise or MA(1 or else. Now, let s use } and } to represent the corresponding error terms of } and } as shown below, and compute and : (1 If we assume, then we can derive the expressions of as: In this case }. Similarly } Thus, both } and } should follow the model. (2 If we assume, where C is a constant, then we can derive the expressions of as: 4
Thus, still }. Same as (1, } and both } and } should follow the model. The general theorem is that (see page 427 of our textbook, for a k-dimensional ARMA(p,q model, the marginal models are ARMA[kp, (k-1p+q]. 3. Derive the h-step forecast and forecast error for VAR(1, mean and covariance of the forecast error. [ ],, ( ( ( ( 5
( ( (( ( ( ( [ ] 4. Eigenvalues and Eigenvectors The equation Ax = y is a linear transformation that maps a given vector x onto a new vector y. Special vectors that map onto multiples of themselves are very important in many applications, because those vectors tend to correspond to preferred modes of behavior represented by the vectors. Such vectors are called eigenvectors (German for proper vectors, and the multiple for a given eigenvector is called its eignevalue. To find eigenvalues and eigenvectors, we start with the definition: Ax x, which can be written as A I x, which has solutions iff det A I The values of that satisfy the above determinant equation are the eigenvalues, and those eigenvalues can then be plugged back into the defining equation Ax x to find the eigenvectors. 6
If 1 and 2 are distinct eigenvalues of a symmetric matrix A, then their corresponding eigenvectors are linearly independent (orthogonal to each other. You will see that eigenvectors are only determined up to an arbitrary factor; choosing the factor is called normalizing the vector. The most common factor to choose is the one that results in the eigenvector having a length of 1. Practice: (1. Find the eigenvalues and eigenvectors of 3 1 A 4 2. 5. Derivation of below: Let s use Σ to represent the variance-covariance matrix of ( as shown Σ ( ( Here, we assume Σ is diagonalizable. If we transform ( into new error terms ( as shown below, we can prove that the variance-covariance matrix Σ of is an identity matrix, which means and are uncorrelated. Σ ( 7
Σ (Σ Σ Σ Σ (Σ Σ Σ Thus, in order to let and be uncorrelated, we just need to find out the transform Σ (. First, we need to find out the eigenvalues and eigenvectors of Σ. By solving the following equation, we can find the eigenvalues of Σ. (Σ ( ( ( ( Thus, one set of eigenvectors can be derived by solving the following equation: (Σ ( ( ( ( ( ( ( ( Σ ( Σ ( 8
( ( ( ( ( Σ Σ ( ( ( ( ( Σ ( If we let and then and Finally, we can derive the expressions for Σ which is Σ ( 9
( ( ( ( ( ( ( ( (Σ 1
6. VAR(1 -- The Reduced Form [ ] Here ( is a sequence of white noise vector where the two elements can be correlated at the same time point. That is, the covariance C may not be zero in: Σ ( ( Note: Here we use Σ to represent the variance-covariance matrix of ( shown above and below. as 7. VAR(1 -- The Structural Form From the reduced form: [ ] Where the covariance C may not be zero in: Σ ( ( By multiplying Σ on both sides, we have the structural form: Σ Σ [ ] Σ Where 11
Σ ( ( That is, the errors terms are entirely uncorrelated, even for the two elements at the same time points. However this also means that for the non-degenerate cases, we have the contemporaneous terms of the other variable as the regressors and in the model as well. The way to convert the VAR from the reduced form to the structural form is not unique. For example, one can also employ the Cholesky Decomposition. See Chapter 8 of our textbook. Briefly, the Cholesky Decomposition (or Cholesky Factorization says that for any positive definite matrix Σ, we can find a lower triangular matrix : [ ] Such that Σ Where is a diagonal matrix. Definition: In linear algebra, a symmetric n n real matrix is said to be positive definite if the scalar is positive for every non-zero column vector of real numbers. The variance-covariance matrix Σ of full-rank qualifies as a positive definite matrix because: Σ Therefore we can transform the reduced form VAR(1 to a structural form as follows: 12
[ ] Where Σ ( ( Note that equivalently, we can also apply the Cholesky Decomposition in the other order to obtain the following structural form: Where and are entirely uncorrelated. Note: There can be various equivalent expressions for the structural forms. Note: The same applies to VAR(p of any dimension. 13