Multivariate Econometrics summary of the Master Econometrics and Operational Research at the VU. This summary includes all course material discussed in lectures and workgroups. The summary includes topics such as: dynamic regression theory, VAR processes, stationarity, ergodicity, lag operators, ma...
• xt is a vector of economic variables (e.g., GDP, interest rates) observed at time t.
• These variables are often interrelated and evolve over time.
{xt : −∞ < t < ∞}
• This sequence is called a ”random sequence” of economic variables.
VAR(1) Model:
E(xt | xt−1 ) = δt + Λxt−1
• δt is a vector of constants (intercepts).
• Λ is an m × m matrix of coefficients showing the influence of past values.
• The model uses the previous time step (xt−1 ) to predict the current value (xt ).
ε t = xt − δt − Λxt−1
• ε t is called the ”mean innovation process” and represents the unpredictable part of
the time series.
Rewritten VAR(1) Model:
xt = δt + Λxt−1 + ε t
Properties of the Error Term (ε t ):
1. The expected value of the error term given past information is zero:
E(ε t | Xt−1 ) = E(xt − δt − Λxt−1 | Xt−1 ) = 0.
2. By the law of iterated expectations, the unconditional expectation of the error term is
zero:
E(ε t ) = 0.
3. The error term is uncorrelated with all lagged values of the variables:
E(ε t x′t− j ) = 0 for all j > 0.
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, Summary Multivariate Econometrics
4. The error term is uncorrelated with its own past values:
E(ε t ε′t− j ) = 0 for all j > 0.
Conditional Distribution of the Error Term:
• To fully specify the data-generating process, we assume a conditional variance for ε t :
E(ε t ε′t | Xt−1 ) = Ω,
where Ω is a constant matrix representing the unconditional variance of ε t .
Gaussian Assumption for Error Term:
• Assume that the conditional distribution of ε t | Xt−1 is Gaussian (normal).
ε t | Xt−1 ∼ MV N (0, Ω),
xt | Xt−1 ∼ MV N (δt + Λxt−1 , Ω),
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Dt (xt | Xt−1 ) = (2π )−m/2 |Ω|−1/2 exp − ε′t Ω−1 ε t .
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• MV N: Multivariate normal distribution.
• Assuming ε t is Gaussian with fixed mean and variance, and uncorrelated over time
implies that ε t is identically and independently distributed (i.i.d).
Reduced Form Representation:
xt = δt + Λxt−1 + ε t
• This is the reduced form of the model, where all right-hand side variables are prede-
termined at time t.
• No variable directly affects other variables at the same time point (no contemporane-
ous effects) which is generally not in line with economic theory.
Structural Form Representation:
Bxt = Γδt + Cxt−1 + ut ,
• B is a full-rank matrix, representing contemporaneous relationships among the vari-
ables.
• B is defined as:
1 b12 ... b1m
.. .. ..
b
. . .
B = 21 ̸= Im
.. .. .. ..
. . . .
bm1 ... ... 1
• The error term ut has properties:
E(ut | Xt−1 ) = 0, E(ut u′t | Xt−1 ) = Σ.
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